ELEMENTARY  TEXT  BOOK 


OF 


Theoretical  Mechanics 

(KINEMATICS  AND  STATICS) 

BY 

GEO.  A.  MERRILL,    B.  S. 

PRINCIPAL   OF 
THE   CALIFORNIA   SCHOOL   OF   MECHANICAL   ARTS 


[AUTHOR'S  EDITION.] 


SAN  FRANCISCO  : 

PRESS    OF    UPTON     BROS. 

I899 


COPYRIGHT,   1899 
BY 

GEO.  A.  MERRILL. 


PREFACE. 


The  subject  of  Mechanics,  as  herein  treated,  is  within  the 
comprehension  of  students  in  the  upper  classes  of  secondary  schools. 
An  intelligent  journeyman,  also,  would  find  no  difficulty  in  reading 
and  understanding  nearly  every  page  of  this  book.  While  algebra, 
geometry  and  trigonometry  are  freely  used  to  facilitate  the 
demonstration  of  principles,  a  person  with  only  a  good  working 
knowledge  of  arithmetic  will  be  able  to  understand  the  principles 
thus  deduced  and  to  apply  them, — especially  by  the  aid  of  the 
graphical  methods  which  are  given  with  some  prominence  through- 
out the  book. 

The  author  feels  that  he  enjoys  almost  a  monopoly  in  this  field, 
so  far  as  American  publications  are  concerned.  The  teaching  of 
Mechanics  as  a  subject  per  se  is  confined  in  the  main  to  collegiate 
courses,  and  the  few  American  text-books  on  the  subject  are  written 
for  students  familiar  with  the  calculus.  A  number  of  more 
elementary  texts  have  been  published  by  English  authors,  but  thtir 
value  to  American  readers  is  impaired  by  the  fact  that  they  are 
usually  compiled  with  a  view  to  meeting  the  conditions  imposed  by 
the  English  examination  system,  which  does  not  conform  to  the 
American  educational  plan. 

The  subject-matter  has  been  restricted  to  Kinematics  and  Statics. 
It  is  left  entirely  for  future  consideration  as  to  whether  or  not  a 
subsequent  volume  on  Kinetics  will  be  issued. 

As  this  is  a  text-book  and  not  a  treatise,  it  is  written  from  the 
standpoint  of  the  student,  without  attempting  to  force  upon  him  any 
rigid  sequence  of  topics  and  ideas  that  a  logical  analysis  of  the 
subject  might  seem  at  times  to  require.  Beyond  an  effort  to  abide  by 
a  few  of  the  fundamental  precepts  of  teaching,  no  one  method  of 
presentation  has  been  used  to  the  exclusion  of  others.  Only  a  few 
experiments  are  required  or  suggested.  Any  good  teacher,  however, 


IV  PREFACE. 

could  easily  arrange  a  parallel  course  of  laboratory  exercises.  In  his 
own  classes  the  author  has  found  that  the  average  student  has 
acquired  from  his  every-day  observations  and  experiences  an 
acquaintance  with  facts  and  phenomena  quite  sufficient  to  enable  him 
to  master  the  subject  without  a  formal  laboratory  course.  It  is 
probably  the  same  in  the  higher  classes  of  all  schools  that  include 
shopwork  and  general  physics  in  their  curricula.  Even  in  the  purely 
academic  high  schools  the  course  of  general  physics  includes,  as  a 
rule,  laboratory  exercises  on  the  simple  machines,  friction,  accelera- 
tion, etc. 

The  present  and  prospective  prominence  of  American  manu- 
facturing industries  not  only  gives  the  average  young  man  a  zeal 
for  the  study  of  Mechanics,  but  it  also  argues  the  need  of 
giving  greater  prominence  to  that  subject  in  the  high  schools. 
Looked  at  from  almost  any  standpoint,  the  few  pages  usually  alloted 
to  Mechanics  in  the  elementary  text-books  of  Physics  are  grossly 
inadequate.  Doubtless  the  teaching  of  Mechanics  as  a  separate 
subject  would  be  stimulated  by  the  publication  of  a  number  of  good 
text-books  in  that  line. 

The  author's  thanks  are  extended  to  Dr.  Caroline  Baldwin 
Morrison  for  numerous  suggestions  and  for  the  careful  reading  of  the 
press-proofs. 

GEO.  A.  MERRILL. 
SAN  FRANCISCO,  FEB.,  1899. 


CONTENTS. 


INTRODUCTION. 

Fundamental  Branches — Kinematics,  Statics, 
Kinetics.  Fundamental  Ideas— Space,  Mass, 
Time. . 


SECTION  I— KINEMATICS. 


1-2 


CHAPTER  I.       MOTION.     VELOCITY. 

Velocity,  Uniform  Motion,  Average  Veloc- 
ity, Relative  Motion,  Graphical  Representa- 
tion of  Motion 3-7 

CHAPTER  II.     COMPOSITION  OF  VELOCITIES. 

Parallel  Motions,  Components  at  Right 
Angles,  Components  at  Any  Angle,  Resolving 
a  Velocity  into  Components,  Triangle  of 
Velocities  and  Polygon  of  Velocities,  Result- 
ant for  Angles  Greater  than  90° 8-24 

CHAPTER  III.    CIRCULAR  MOTION. 

Motion  of  a  Body  in  a  Circle,  Angular 
Velocity,  Composition  of  Circular  and  Rec- 
tilinear Motion,  Composition  of  Two  Circular 
Motions,  Cycloids 25~37 

CHAPTER  IV.    ACCELERATION. 

Idea  of  Acceleration,  Definition  of  Ac- 
celeration, Formulae  for  Uniformly  Accelerated 
Motion,  Acceleration  of  Gravity. 

Use  of  Co-ordinate  Axes,  Graphical  Repre- 
sentation of  Distance  Traveled  by  an  Acceler- 
ated Body. 

Projectiles,  Elevation  and  Range,  Maxi- 
mum Range 38-56 


VI 


CONTENTS. 


SECTION  II— STATICS. 


CHAPTER  I. 


CHAPTER  II. 


CHAPTER  III. 


CHAPTER  IV. 


FORCE.    MASS. 

Different  Kinds  of  Force,  Definition  of 
a  Force,  Resistance,  Statical  Conditions, 
Action  and  Reaction,  Composition  and 
Resolution  of  Forces,  Law  of  Gravitation, 
Mathematical  Expression  of  the  Law  of 
Gravitation,  Weight  of  a  Body  Beneath  the 
Earth's  Surface,  Mass,  Density,  Heavy  and 
Light  Bodies  Fall  at  the  Same  Rate 57-73 

WORK.    POWER.    ENERGY. 

Foot-Pound,  Horse-Power,  the  Watt, 
Energy,  Potential  Energy,  Kinetic  Energy, 
Molecular  Energy  and  Mechanical  Energy, 
Chemical  Energy,  Transference  and  Trans- 
formations of  Energy,  Conservation  of  Energy, 
Graphical  Representation  of  Work,  Indicator 
Diagram 74~93 


CENTER  OF  GRAVITY. 

Center  of  Figure,  Center  of  Mass,  Cen- 
ter of  Gravity,  To  Locate  the  Center  of 
Gravity  of  a  Body,  Equilibrium,  Stability  of 
Equilibrium,  Degree  of  Stability 94-108 

PRINCIPLES  OF  MACHINES.    THE  LEVER. 

Tools  and  Machines,  Efficiency  of  Ma- 
chines, the  Simple  Machines  or  Mechanical 
Powers,  the  Lever,  Principle  of  Virtual  Work, 
Principle  of  Moments,  Moment  Due  to  Weight 
of  Lever,  Three  Kinds  of  Levers,  Compound 
Levers,  Safety  Valves,  Pressure  on  the  Ful- 
crum, Parallel  Forces,  the  Couple,  Bent 
Levers,  Moment  of  a  Force  Acting  Obliquely 
on  a  Lever,  Mechanical  Advantage 109-131 


CONTENTS. 


Vll 


CHAPTER  V.      MACHINKS. 

Wheel-and-Axle,  Gearing  and  Shafting, 
the  Pulley,  Fixed  Pulley,  Movable  Pulley, 
Combinations  of  Pulleys,  the  Inclined  Plane, 
the  Wedge,  the  Screw,  Endles,s  Screw,  the 
Cam  and  Eccentric,  the  Toggle-joint,  Differ- 
ential Motion,  Compound  Machines 132-164 

CHAPTER  VI.    FRICTION. 

Sliding  Friction,  Coefficient  of  Friction, 
Static  Friction  and  Kinetic  Friction,  Friction 
Always  a  Resistance,  Determination  of  the 
Coefficient  of  Friction  by  Means  of  the  L/imit- 
ing  Angle  or  Angle  of  Repose,  Work  Done  in 
Dragging  a  Body  by  Sliding,  Rolling  Friction, 
Work  Done  in  Dragging  Vehicles,  Anti- 
Friction  Wheels  and  Ball  Bearings,  Lubri- 
cated Surfaces,  Friction  of  Ropes  Belts  and 
Cables,  Measurement  of  Power  Transmitted 
by  Belts,  Methods  of  Increasing  the  Efficiency 
of  Belts 165-195 


THEORETICAL  MECHANICS. 


INTRODUCTION. 


The  scope  of  Theoretical  Mechanics  is  indicated  in  the 
following  outline  of  its  fundamental  branches : 

I.  The    branch   called   Kinematics   is   a   study   of  the 
motions   of  bodies  without  reference  to  the  cause  of  their 
motion  ;  it  inquires  how,  or  in  what  manner,  a  body  moves, 
as  distinguished  from  why  it  moves. 

II.  The  branch  called  Statics  is  a  study  of  pressures, 
etc.,     in   bodies  ; — that  is,  the  influe.n££_-X)£l-ibrces^  that  are 
unable   to   move^  the   body   acted  upon,   because   of  other 
balancing  forces  or  resistances.    When  an  internal  resistance 
balances   an  exterior  force  the  body  is  said  to  be  under 
'  'stress." 

Pressures  in  fluids  give  rise  to  special  considerations, 
which  are  grouped  under  a  subdivision  of  Statics  called 
Hydrostatics. 

In  the  study  of  Statics,  geometric  and  graphic  methods 
are  applied  so  widely,  as  in  computations  pertaining  to  roof 
trusses,  bridges,  etc.,  that  it  has  been  found  convenient  to 
give  this  mode  of  treating  the  subject  a  special  name, 
Graphostaties. 

III.  The  third  branch  deals  with  the  relation  between 
the  motion  of  a  body  and  the  force  or  forces  producing  the 
motion,  and  is  called  Kinetics.       It  considers  why  the  body 
moves,    and   the  relation  between  the  manner  in  which    it 
moves  and  the  influences  causing  the  motion. 


2  THEORETICAL   MECHANICS. 

It  will  be  observed  that  in  Statics  and  Kinetics  we  have 
to  deal  with  forces.  When  classed  together  on  that  basis 
these  two  branches  are  sometimes  called  Dynamics,  in 
contrast  with  Kinematics,  or  the  study  of  pure  motion. 


Fundamental  Ideas.  The  principles  of  mechanics  are 
deduced  and  developed  by  mathematical  operations,  based 
upon  the  fundamental  ideas  of  space,  time  and  mass.  The 
mathematical  operations  used  in  an  elementary  treatment  of 
the  subject  are  mainly  arithmetical,  but  are  greatly  facilitated 
by  the  use  of  Algebra,  Geometry,  and  Trigonometry.  The 
ideas  of  space,  time,  and  mass  are  usually  presented  in  the 
earlier  parts  of  an  elementary  course  of  physics. 

Measurements  of  length,  area,  volume  and  capacity,  and 
also  angular  measurements,  are  all  comprehended  in  the 
idea  of  Space. 

Although -a  rigid  definition  of  Mass  is  rarely  required 
outside  the  subject  of  mechanics,  the  related  ideas  of  weight, 
density,  and  specific  gravity  are  frequently  met  with. 

The  unit  of  Time,  a  second,  is  well  understood  as"  meaning 
a  certain  fraction  of  a  day,  but  it  should  be  remembered 
that  a  "day"  is  not  always  of  fixed  length.  If  the  earth 
had  but  one  motion,  a  rotation  upon  its  axis,  the  successive 
transits  of  the  sun  across  any  meridian  (noonday)  would 
always  occur  at  equal  intervals.  But  since  the  earth  also 
revolves  around  the  sun,  and  in  an  elliptical  rather  than  a 
circular  orbit,  the  interval  between  transits  is  not  constant. 
For  scientific  accuracy  this  difference  must  sometimes  be 
allowed  for  ;  but  for  most  purposes  it  is  sufficient  to  take  as 
the  length  of  a  "mean  solar  day"  the  average  of  these 
successive  intervals  of  transit  of  the  sun  across  any  meridian. 
The  ordinary  second  is  60x^)x24  °^  *his  mean  solar  day. 


SECTION  L 
KINEMATICS. 


CHAPTER  I. 
MOTION.      VELOCITY. 

Velocity.  When  a  body  starts  to  move  from  a  given 
point  a  full  description  of  its  motion  involves  : 

1 .  Its  direction  of  motion  ; 

2.  Its  rate  of  motion. 

The  idea  of  direction  is  derived  from  the  conception  of 
space,  and  the  "  rate  of  motion  "  implies,  in  turn,  the  other 
two  fundamental  ideas, — distance  and  time.  By  the  rate  of 
motion  we  mean  the  distance  traveled  in  a  unit  of  time.  This 
is  usually  called  the  Velocity  of  the  body.* 

Examples : 

1.  (a)    A  velocity  of  10  meters  per  sec.   is  equivalent  to  hoiv 
many  feet  per  second  f 

(b}    Express  the  same  velocity  in  feet  per  minute. 

2.  (a]     Express  in  feet  per  sec.  a  velocity  of  too   meters  per 
minute. 

(b)     Express  the  same  velocity  in  feet  per  hour. 


*    The  velocity  of  a  body,  when  considered  independently  of  the  direction  of 
motion,  is  sometimes  called  the  speed  of  the  body. 


4  THEORETICAL   MECHANICS. 

3.  (a]     A  velocity  of  1000  meters  per  min.  is  equivalent  to  how 
many  miles  per  hour  f 

(/>)     Express  the  same  velocity  in  feet  per  sec.  f 

4.  Which  velocity  is  the  greater  and  by  hoiv  much, — 40  miles  per 
hour,  or  12  meters  per  sec.  ? 


Uniform  Motion;  Average  Velocity.  The  distance 
traveled  in  any  given  time  by  a  body  moving  with  uniform 
velocity  is  expressed  by  the  formula 


(i) 


By  transformation  this  formula  becomes 

d 


(2) 


which  indicates  that  the  velocity  of  any  moving  body  may 
be  found  by  dividing  the  distance  traveled  by  the  units  of 
time  consumed.  (Compare  this  statement  with  the  defini- 
tion of  velocity,  already  given.)  This  is  true  even  if  the 
speed  of  the  body  is  not  uniform,  the  quotient  dft  represent- 
ing in  this  case  the  average  velocity  during  the  time  under 
consideration. 

Example : 

A  train  travels  10  miles  at  a  rate  of  20  miles  per  hour  ;  then  4  miles 
at  an  average  rate  of  30  miles  per  hour ;  then  6  miles  at  a  uniform 
rate  of  40  miles  per  hour  ;  then  it  comes  to  the  rest  after  traveting  a 
distance  of  I  mile  {slowing  down],  running  meanwhile  at  an  average 
rate  of  20  miles  per  hour ;  it  stands  for  7  minutes  ;  then  it  starts  and 
runs  for  20  minutes  at  the  average  speed  of  21  miles  per  hour.  What 
has  been  its  average  velocity  for  the  entire  time  ? 


MOTION.         VELOCITY.  5 

Relative  Motion.  An  object  at  rest  on  the  earth's  surface 
may  or  may  not  be  in  a  condition  of  absolute  rest.  It  moves 
with  the  earth  in  diurnal  rotation,  and  it  progresses  with  the 
earth  in  its  path  of  revolution  around  the  sun,  but  there  may 
be  other  motions  due  to  the  sweeping  of  the  solar  system 
through  space,  the  effect  of  which  may  be  to  bring  the  object 
under  consideration  to  a  condition  of  absolute  rest  for  the 
instant.  More  probably,  these  unknown  influences  add  to 
the  complexity  of  the  actual  motion,  but  relatively  to  the 
earth  the  body  continues  at  rest. 

Examples  : 

1.  Two  trains  pass  each  other  at  a  station,  called  A,  One 
train,  called  E,  is  moving  eastward  at  the  rate  0/30  miles  per  hour  ; 
the  other,  W,  is  moving  westward  25  miles  per  hour. 

Find: — (a]  The  velocity  of  E  relative  to  A. 

(b]  "          "  "    W       "        "  A. 

(c]  "          "  "    E         "        "   W. 

(d]  "          "  "     W        "        "  E. 

DISCUSSION  : — In  this  example,  if  we  consider  only  the  rate  of 
motion,  then,  since  the  train  E  would  be  located  30  miles  from  the 
station  at  the  end  of  an  hour,  it  would  be  sufficient  to  say  that  its 
velocity  relatively  to  the  station  is  30  miles  per  hour.  But,  if  we  also 
consider  direction  of  motion,  then  we  must  add  that  the  motion  of 
the  train  E  relatively  to  the  station  is  in  an  Eastward  direction. 

Likewise  the  motion  of  W  relatively  to  A  is  25  miles  per  hour 
Westward. 

At  the  end  of  the  hour  the  train  E  is  55  miles  East  of  the  train  W. 
Without  giving  to  the  train  W  any  credit  for  its  part  of  the  transaction, 
we  simply  recognize  the  fact  that  in  the  hour  of  time  E  reaches  a 
position  of  55  miles  East  of  W,  and  we  express  this  by  saying  that  the 
velocity  of  E  relatively  to  W  has  been  55  miles  per  hour  Eastward. 

Conversely,  we  must  say  of  W,  that  its  velocity  relatively  to  E 
has  been  55  miles  per  hour  Westward. 


6  THEORETICAL    MECHANICS. 

2.     A   train  400  feet  long  travels  between  two  mile-posts  at  a 
uniform  rate  of  30  miles  per  hour. 


CD 


At  position  \  1 1  a  person  starts  from  the  last  car  to  walk  through 
the  train,  and  reaches  the  front  at  the  instant  the  train  reaches  |T|. 

(a)  With  what  average  velocity  in  miles  per  hour  did  he  walk  ? 

(b)  What  was  his  average  velocity  relatively  to  the  ground? 

(c)  What  would  have  been  the  answers  to  (a)  and  (b}  if  he  had 
walked  to  the  back  from  the  front  of  the  train  ? 


Graphical  Representation  of  Motion.  The  motion  of  a 
body  at  any  instant, — both  the  magnitude  and  the  direction 
of  its  velocity, — can  be  represented  by  a  straight  line.  For 
this  purpose  we  can  adopt  any  convenient  scale  of  magnitude 
and  any  arbitrary  notation  to  indicate  direction.  For 
instance,  a  velocity  of  20  miles  an  hour  in  an  eastward 
direction  could  be  represented  by  a  line  2  inches  long, 
(i  inch=io  miles  per  hour),  drawn  horizontally  from  left 
to  right  from  the  starting  point,  the  direction  being 
indicated  by  an  arrow,  as  OA  in  the  following  figure : 


If  in  connection  with  this  motion  we  wish  to  refer  to  a 
second  motion, — say  a  velocity  of  15  miles  an  hour  northward 
from  the  same  point,  — we  would  have  to  adhere  to  the  same 
scale,  (i  inch  =  10  miles  per  hourj,  and  instinctively  we 
would  draw  this  line  vertically  upward. 


MOTION.         VELOCITY.  7 

Exercises : 

1.  Using   the  same  notation  represent  each    of  the  following 
motions,  using  the  same  starting  point  for  all : 

22    miles  per  hour  Southward. 

j  "  "  •'     Westward. 

17  «  "  "     N.  E. 

14.5  "  "  "     S.  E. 

3  "  "  "     63°  S.  of  E. 

20.5  «•  «  '77°  N.ofE. 

77.7  "  "  "    14°   W.  of  N. 

II?       „         „  „     ^    Wm  ofS^ 

It  should  be  carefully  noted  that  the  scale  of  magnitude  in  this 
figure  is  not  a  scale  of  distance,  and  the  lines  themselves  do  not 
represent  distances  merely,  but  velocities.  A  similar  method  could 
be,  and  frequently  is,  used  for  distances,  and  hence  it  becomes 
necessary  to  keep  clearly  in  mind  the  real  signification  of  each 
diagram,  and  each  line  in  it.  Take  for  example,  the  confusion  that 
might  arise  in  interpreting  a  diagram  constructed  as  follows  : 

2.  On  a  scale  of  i  inch  — 10  meters,  construct  a  diagram  to  represent 
the  PATH  of  a  body  moving  as  follows  :     From  the  starting  point  the 
body  moves  westward  10  seconds,  at  the  rate  0/3  meters  per  second; 
thence  northward  2.4  seconds  at  the  rate  of  5  meters  per  second;  thence 
north-east  7  seconds,  at  the  rate  of  7.7  meters  per  second;  thence  30° 
east  of  south  for  6  seconds  at  the  rate  of  2  meters  per  second. 

Having  constructed  this  diagram  we  can  use  it  for  any  measure- 
ment of  distances  concerned  with  the  path  of  this  body,  but  if  we  wish 
to  consider  incidental  questions  of  velocity  it  becomes  necessary  to 
introduce  additional  computations  involving  the  consideration  of  time. 
For  instance,  at  the  end  of  the  motion  how  far  was  this  body  from  the 
starting  point?  And  if  a  second  body  had  moved  in  a  straight  line 
from  the  starting  point  to  finish,  instead  of  by  the  roundabout  path, 
at  what  rate  should  it  have  traveled  in  order  to  arrive  at  the  terminal 
point  simultaneously  with  the  first  body  ? 

It  is  true  that  a  line  which  represents  a  velocity  may  be 
regarded  at  the  same  time  as  representing  distance,  provided 
we  remember  that  it  is  distance  per  unit  of  time. 


CHAPTER  II. 
COMPOSITION  OF  VELOCITIES. 

Sometimes,  and  generally,  the  actual  motion  of  a 
body  is  the  result  of  several  different  influences  acting 
simultaneously,  as  in  rowing  a  boat  in  a  tidal  current. 
Heretofore,  in  speaking  of  the  velocity  of  a  body  we  have 
thought  of  it  only  as  a  simple  motion  in  one  direction,  taking 
it  as  we  find  it,  no  matter  how  many  different  influences  may 
have  contributed  to  give  the  body  this  velocity  and  path. 
When  two  or  more  influences  act  simultaneously  to  produce 
motion  in  a  body  each  influence  has  its  full  effect  in  its  own 
direction,  as  if  the  other  influences  did  not  exist.  The 
motion  that  each  of  these  influences  would  produce  is  called 
a  component,  and  the  actual  motion,  as  we  have  been  con- 
sidering it,  is  called  the  resultant  of  all  these  components. 
There  are  times  when  the  component  motions  are  known  and 
the  resultant  cannot  be  found  except  by  computing  it  from 
the  components  ;  at  other  times  the  actual  motion  is  given 
and  it  is  desired  to  find  its  component  parts.  We  have 
only  to  take  the  literal  meaning  of  l '  component ' '  to  see 
that  the  resultant  is  obtained  by  "  putting  together, " 
or  combining,  these  more  elementary  motions,  but  the 
mathematical  process  of  doing  this  is  not  always  simple, 
although  sometimes  it  is  a  mere  addition  or  subtraction. 

For  the  convenience  of  the  beginner  it  is  well  to  assume 
three  cases,  exemplified  as  follows  : 

(i)  If  a  person  walks  directly  forward  through  a  moving 
car  his  resultant  is  his  rate  of  walking  plus  the  velocity  of 
the  train.  This  would  be  the  composition  of  motions 
parallel  to  each  other. 


COMPOSITION   OF   VELOCITIES.  9 

(ii)  If  he  walks  straight  across  the  car  his  resultant 
velocity  is  the  hypothenuse  of  a  right  triangle,  of  which  one 
side  represents  the  rate  at  which  he  walks  and  the  other  side 
represents  on  the  same  scale  the  velocity  of  the  train.  This 
is  illustrated  in  Fig.  3.  The  man  starts  from  A  to  move 
across  the  car  toward  the  window  W.  Suppose  that  in  a 
unit  time  he  covers  that  part  of  the  distance  represented  by 
AB :  then  the  arrow  AB  represents  his  velocity  of  walking. 


Now,  since  the  car  is  in  motion,  suppose  that  it  progressed 
from  position  I  to  position  II  while  the  man  walked  from 
A  to  B, — that  is,  in  a  unit  of  time.  Then  the  arrow  BB' 
will  represent  the  velocity  of  the  car, — or  any  point  in  it,  as 
B.  Hence,  at  the  end  of  the  unit  of  time  the  man  finds 
himself,  not  at  B,  (as  he  would  if  the  car  were  at  rest),  but 
at  B' ,  and  the  actual  path  he  has  traveled  in  space, — or, 
better,  relatively  to  the  ground, — is  AB' .  This  hypothenuse, 
therefore,  represents  his  resultant  velocity,  on  the  same  scale 
2,$  AB. 

Properly  speaking,  the  two  components  of  AB'  are  the 
two  motions  of  the  man  himself, — one  AB,  relatively  to 
the  car,  (which,  also,  would  have  been  his  only  motion 
relatively  to  the  ground  if  the  train  had  not  been  moving 
when  he  walked  across)  ;  and  the  other  AA\  which  would 
have  been  his  only  motion  relatively  to  the  ground  had  he 


10 


THEORETICAL    MECHANICS. 


stayed  in  his  seat  while  the  train  moved.  These  two 
motions  occurring  conjointly,  his  resultant  has  a  direction 
between  the  two.  In  finding  the  magnitude  and  direction 
of  AB'  it  makes  no  difference  whether  we  use  the  triangle 
ABB\  or  its  equal  AA'B'\  each  is  half  the  rectangle 
ABB'A'. 

(iii)  If  he  walks  obliquely  across  the  car  towards  a 
point  X,  as  in  Fig.  4,  his  resultant  motion  relatively  to 
the  ground  will  be  the  side  AB'  of  the  obtuse  triangle 
ABB'.  This  case  differs  from  the  last  only  in  the  oblique- 
ness of  the  two  components  AB  and  A  A',  but  this  difference 


Fig.  4. 

is  a  very  important  one  from  a  mathematical  point  of  view. 
It  is  only  when  the  angle  BAA'  is  90°,  or  45°,  or  60°,  or 
120°,  that  the  triangle  ABB',  or  AA'B,  can  be  readily 
solved  without  using  trigonometry. 

Composition   of  two   motions   parallel   to   each   other. 

In  Kxatnple  2,  page  6,  as  the  man  walks  through  the  train,  he  is 
subject  to  the  additional  motion  of  the  train  itself.  In  part  (a)  of  the 
problem  he  covers  in  two  minutes  a  distance  of  400  feet  (by  walking) 
+5280  feet  (by  the  progress  of  the  train).  These  two  component 
motions  take  place  simultaneously,  and  since  they  are  in  the  same 
direction  the  resultant  velocity  of  the  man  relatively  to  the  ground 
is  the  sum  of  the  two  components;  v  —  dlt  —  5280/-4J)-°  feet  per 
minute. 


COMPOSITION   OF   VELOCITIES.  II 

Observe  that  the  two  motions  occur  in  the  same  interval  of  time  ; 
if  they  had  been  consecutive,  instead  of  concurrent,  the  conditions 
would  have  been  very  different  and  would  not  have  come  within  the 
scope  of  "composition  of  velocities."  That  is,  if  the  man  had  remained 
in  his  seat  at  the  rear  of  the  train  until  the  train  came  to  rest  at  mile- 
post  \~2~\t  he  would  have  covered  a  distance  of  only  5280  feet  in  the  two 
minutes  ;  if,  thereafter,  he  had  walked  to  the  front  of  the  train  while  it 
was  still  in  waiting  at  mile-post  [2],  he  would  have  required  two  minutes 
additional  time  in  which  to  walk  the  length  of  the  train,  400  feet. 
His  velocity  during  the  first  two  minutes  would  have  been^2j8a  feet 
per  minute,  and  during  the  second  two  minutes,  ^f-0  feet  per  minute. 
If  there  were  any  need  to  know  his  average  velocity  during  the  total 
of  these  two  intervals  of  time  it  would  be  ^-S-^-LQO  fee^  per  minute, 
but  his  motion  during  the  second  two  minutes  has  no  effect  to  change 
the  rate  at  which  he  covered  ground  during  the  first  two  minutes,  and 
vice  versa.  In  other  words  the  "composition  of  velocities"  is 
concerned  only  with  the  resultant  of  two  or  more  components,  acting 
conjointly  or  simultaneously. 

The  resultant  of  two  or  more  parallel  motions  in  the 
same  direction  is  equal  to  the  sum  of  the  components  ;  if  the 
two  motions  are  in  opposite  directions  their  resultant  is 
equal  to  their  difference.  Using  the  signs  +  and  —  to 
indicate  opposite  directions,  we  can  say,  in  general  terms, 
that  the  resultant  of  two  parallel  motions  is  equal  to  their 
algebraic  sum. 

A  person  rowing  directly  with  or  against  wind  and  tide 
is  subject  to  three  influences,  each  of  which  if  undisturbed, 
would  cause  him  to  move  with  a  certain  velocity.  His 
resultant  or  actual  motion  could  be  found  by  summing  up 
the  three  motions  that  would  have  been  occasioned  by  these 
influences  acting  singly. 

Examples  : 

1.  In  Example  <?,  page  5,  if  the  man  had  walked  to  the  back  of  the 
train  from  the  front,  as  in  part    "r,"  what    would  have  been  his 
resultant  velocity  ? 

2.  In  Example  2,  page  7,  were  the  motions  designated  as  simulta- 
neous.  or  as  consecutive  ? 


12 


THEORETICAL   MECHANICS. 


Components  at  Right  Angles.      The  statements  in  (ii) 
page  9,  will  aid  in  the  solution  of  the  following 
Examples : 

1.     If  a  man  undertakes  to  row  straight  across  a  channel  in 
which  there  is  a  current,  his  course  will  be  oblique. 

Let  AB  =  his  velocity  of  rowing, 
and  A  C  =  his  velocity  with  the  current. 
Then  the  resultant  AD,  the  diagonal  of  the 
parallelogram,  will  be  the  direction  of  his  course. 

(a)  If  AB  =  4  miles  per  hour,  and  A  C  =  j 
r/iiles  per  hour,  what  is  his  resultant  velocity  ? 

(b )  If  the  DISTANCE  straight  across  the  channel 
(AE}  is  8  miles,  what  is  the  distance  AF  ? 

How  long  does  it  take  him  to  reach  F  ? 
How  long  would  it  have  taken  him  to  reach 
E,  if  there  had  been  no  current  ? 

(c}  Find  the  value  of  the  angle  DAB,  by 
trigonometry. 

2.  If  he    wants   to  go  straight  across  he 
must  head  the  boat  up  stream,  as  AB'. 

Let  AC'  —  j  miles  per  hour,  as  before,  and  let 
the  DISTANCE  AE  =  S  miles. 

In  what  direction  and  with  what  velocity 
must  he  row  in  order  to  reach  E,  directly 
opposite  A,  in  2  hours  ? 

3.  A  boat  B,  joo  yards  from  shore  and  100 
)||i   yards   up   stream  from    A,    is  carried   down 

Fig.  6.  stream  j>  miles  per  hour. 

(a)  If  it  is  kept  headed  perpendicular 
to  the  shore  with  what  velocity  must  it 
be  rowed  in  order  to  land  at  A  f 

(£)  What  will  be  the  resultant 
velocity  ? 

(c)     What  is  the  magnitude  of  the 
angle  ABC? 
Fig-  7- 


COMPOSITION    OF   VELOCITIES.  13 

4.     A  steamship,  S,  6  miles  from  the  shore,  AB,  is  stipposed  by 
those  on  board  to  be  sailing  from  south  to  north  10  miles  per  hour,  but 
really  drifts  towards  shore  at  the  rate  of  2  miles 
per  hour.     The  shore  extends  north  and  south. 

(a]     How  long  before  the  steamship  will  reach 
the  shore  ?    Designate  the  point  at  which   it  will 


i 

A1 
Fig.  8. 


strike.     With  what  velocity  will  it  strike  ? 

(b}  If  it  is  desired  that  the  steamship  shall 
progress  exactly  northward  at  the  rate  of  10  miles 
Per  hour,  in  what  direction  and  with  what  velocity 
must  it  steam  in  order  that  the  action  of  its  engines 
combined  with  the  influence  of  the  current  may 
give  it  the  desired  resultant  motion  ? 


Note  how  these  examples  illustrate  the  statement,  previously 
made,  that  when  two  or  more  motions  act  simultaneously  to  produce 
motion  in  a  body,  each  influence  has  its  full  effect  in  its  own 
direction,  as  if  the  other  influences  did  not  exist.  In  Example  4(a] 
it  will  be  noticed  that  the  ship  drifts  towards  the  shore  at  a  certain 
fixed  rate,  no  matter  how  fast  it  is  sailing  in  a  direction  parallel  to 
the  shore;  and  conversely,  its  progress  from  south  to  north  is  not 
influenced  by  the  drifting  shoreward.  Likewise,  in  Kxample  i  the 
current  does  not  retard  the  man's  progress  across  the  channel,  but 
simply  carries  him  down  stream.  - 


Two  Components  at  Any  Angle.  This  is  the  general 
case  and  requires  the  use  of  trigonometry  for  solution, 
although  results  sufficiently  reliable  for  most  purposes  can 
be  obtained  by  graphical  methods  from  carefully  constructed 
diagrams. 

Suppose,  in  a  given  case,  that  a  body  is  subject  to  two 
simultaneous  motions,  one  120  feet  per  second  and  the  other  70  feet 
per  second,  the  directions  of  the  two  differing  by  60°.  For  graphical 
representation  assume  a  scale  of  i  inch  =  100  feet  per  second.  L,et  O 
be  the  starting  point  of  the  body.  Draw  OA  equal  to  1.2  inches,  to 
represent  one  of  the  velocities,  and  OB,  equal  to  0.7  inch,  to 


14  THEORETICAL    MECHANICS. 

represent  the  other,  the  angle  or  difference  of  direction  between 

them  being  made  equal  to 
60°.  From  point  A  as  a 
center  describe  an  arc  whose 
radius  is  equal  to  OB,  and 
from  B  as  a  center  with 
radius  equal  to  OA  describe 
a  second  arc  intersecting  the 
first  at  C.  The  figure  OA  CB 
Fig-  9-  is  thus  a  parallelogram.  Its 

diagonal  OC  represents  the 

resultant  of  OA  and  OB  in  both  magnitude  and  direction,  on  a  scale 

of  i  inch=  100  feet  per  second. 

Calling  these  components  a  and  b  respectively  and  the 
resultant  r,  it  can  be  shown  by  trigonometry  that 


cosO,  (3)* 

where  0  is  the  angle  between  a  and  b, 

It  can   also  be  shown  that  this   formula   is   the   same 
whether  0  be  acute  or  obtuse. 

Examples : 

1.  A  boat  sailing  westward  ij  miles  per  hour  is  carried  by  the 
tide  in  a  direction  23°  W.  of  S.  at  the  rate  0/3.5  miles  per  hour. 

Find  its  resultant  motion  (velocity  and  direction]. 

2.  A  bird  flying  in  a  N.  E.  direction  at  the  rate  0/28  miles  per 
hour  encounters  a  wind  blowing  from  a  direction  12°  W.  of  N.  which 
carries  him  out  of  his  course  at  the  rate  of  19  miles  per  hour.     Find 
his  resultant  velocity  and  direction. 


3.     In  the  formula  r  =  ]    a'2+b2-f-2ab  cos  6,    if  0=i8o°  what  is 
the  value  of  r  f      What  is  the  value  of  r  when  8  =  o°?     When,  6  =  po°  f 


*  It  should  be  carefully  noted  why  this  formula  differs  from  the  formula  for 
the  trigonometric  solution  of  a  triangle,  given  two  sides  and  their  included  angle. 


COMPOSITION    OF   VELOCITIES. 


Resolving  a  Velocity  into  Components. 

Example : 

If  a  body  is  moving  in  a  direction  exactly  N.  W. ,  ^.j  miles  per 
hour,  how  fast  is  it  progressing  northward,  and  how  fast  westward? 
Solve  this  by  constructing  a  diagram  to  scale,  and  also  by 
trigonometric  computation. 

In  this  instance  we  have  assumed  the  given  velocity  to 
be  a  resultant,  and  have  resolved  it  into  two  components, 
one  north  and  the  other  west.  We  might  have  resolved  it 
into  other  components, — two  components  in  directions  other 
than  north  and  west,  or  even  into  several  components. 


m 


is: 


Fig.  10. 

Starting  with  the  given  velocity  of  4.5  miles  per  hour  in 
a  N.  W.  direction,  we  can  draw  an  infinite  number  of 
parallelograms  of  which  this  line  will  be  a  diagonal.  In 
Fig.  10,  the  two  sides,  a  and  b,  of  any  one  of  the  parallelograms 
may  be  taken  as  the  two  components,  of  which  the  actual 
motion  of  4.5  miles  an  hour  in  a  N.  W.  direction  is  the 
resultant.  Sometimes,  as  in  the  case  of  a  ship  sailing  in  an 
invisible  current,  it  is  utterly  impossible  to  discover  the  true 
components  which  have  operated  to  give  the  resultant 


1 6  THEORETICAL    MECHANICS. 

motion,  but  as  a  rule  sufficient  conditions  are  known  from 
which  to  determine  what  the  elementary  motions  would 
have  been  of  each  if  the  component  influences  had  acted 
singly. 

To  resolve  a  resultant  into  two  components  requires  the 
solution  of  a  triangle,  (as  in  the  converse  proposition  of 
finding  the  resultant  from  the  components),  and  hence  at 
least  three  parts  of  the  triangle  must  be  known.  If  we  start 
with  the  resultant  given,  the  conditions  of  the  problem  must 
state,  or  imply,  (i)  the  lengths  of  the  other  two  sides;  or  (2), 
one  of  these  sides  and  an  angle;  or  (3)  two  of  the  angles. 

The  resolution  of  a  velocity  into  components  was 
illustrated  in  Examples  2  and  3,  and  the  last  part  of 
Example  4,  pp.  12  and  13. 

This  triangle  to  be  solved  always  contains  one  of  the  components 
and  a  side  parallel  and  equal  to  the  other  component,  but  it  never 
contains  both  components.  This  fact  frequently  occasions  confusion 
in  the  graphical  representations  that  accompany  this  branch  of 
kinematics,  and  leads  beginners  into  many  difficulties  in  gaining  a 
clear  conception  of  the  idea  of  the  relation  between  a  resultant  and 
its  components.  Remember  that  the  resultant  is  always  the  diagonal 
of  a  parallelogram  of  which  the  components  are  two  sides;  and  that 
the  two  components  and  the  resultant  all  start  from  the  same  point. 

Exercises : 

Solve  by  trigonometry.  Also,  as  a  check  upon  errors,  construct 
diagram  on  convenient  scale  in  each  case.  Those  who  have  not 
studied  trigonometry  can  get  sufficiently  reliable  results  from  diagrams. 

1.  A  ship  is  sailing  at  the  rate  of  10  miles  per  hour  and  a  sailor 
climbs  the  mast  200  feet  high  in^jo  seconds,     find  his  velocity  relative 
to  the  earth. 

2.  A  balloon  leaving  the  ground  ascends  with  a  vertical  velocity 
of  70  feet  per  second  and  is  carried  with  the  wind.     If  it  rises  at  an 
angle  ofSf  with  the  horizon,  at  what  rate  is  it  moved  by  the  wind? 


COMPOSITION   OF   VELOCITIES.  1 7 

3.  One  of  the  rectangular  components  of  a  velocity  of  60  miles  per 
hour  is  a  velocity  of  '30  miles  per  hour  ;  find  the  other  component. 

4.  The  components  in  two  directions  of  a  velocity  of  30  miles  per 
hour  are  velocities  of  ij  and  25  miles  per  hour ;    determine  their 
directions. 

5.  A  steamship  is  headed  in  a  direction  38°  S.  of  E.  in  a  wind 
blowing  from  a  direction  2f  30'  E.  of  N.    If  the  ship's  progress 
proves  to  be  in  a  direction  43°  S.  of  E.  with  a  velocity  of  if  miles  per 
hour,  at  what  rate  was  it  steaming,  and  at  what  rate  was  it  carried 
with  the  wind? 

6.  Find  the  horizontal  and  vertical  components  of  the  following 
velocities : 

(/)     1000  feet  per  second  in  a  direction  inclined  30°  to  the 
horizon. 

(2)  The  same  velocity  in  a  direction  inclined  jjo°  to  the 
horizon. 

(3)  25  miles  per  hour  at  60°  to  the  vertical. 

7.  Find  the  magnitude  and  direction  of   the   resultant  of  the 
following  motions   (four  components'],   to  which  a  body  is  subjected 
simultaneously : 

10  feet  per  second,  E ;    f  feet  per  second,  N ;    13  feet  per 
second,  W ;    and  16  feet  per  second,  S. 

8.  Solve  by  Geometry  : 

(a)  Given  two  equal  components  at  an  angle  of  p=6o°. 
Find  the  resultant.  (Hint :  The  diagonals  of  a  rhombus 
bisect  each  other  at  right  angles.) 

(b}     Find  the  resultant  of  two  unequal  components, 
p.  "  a  and  b,  at  an  angle  of  60°. 

(c)  Find  the  resultant  of  two  equal  components  at  an  angle 

Of  120°. 

(d]  Find  the  resultant  of  two  unequal  components  at  an 
angle  of  120°, 

9.  In  formula  3,  p.  14,  substitute  for  6  the  above   values,  60° 
and  120°,  and  compare  with  the  geometric  results. 


1 8  THEORETICAL  MECHANICS. 

10.  The  Ship  Problem.  A  ship  is  pointed  W.  to  £.,  with  her 
mainsail  set  at  an  angle  of  2C?  (5  in  diagram],  Fig.  n-a.  The  wind 
blows  at  an  angle  of  75*  with  the  sail  (indicated  by  ft  in  diagram}, 
with  a  'velocity  0/39  miles  per  hour. 

(a]  What  is  the  component  of  wind  velocity  in  the  direction  of 
the  ship? 

DISCUSSION: — This  question  is  not  intended  to  imply  that  the 
motion  of  a  ship  is  caused  to  any  great  extent  by  the  component  of 
the  wind  velocity  parallel  to  the  direction  in  which  the  ship  is  headed. 
The  main  motive  power  comes  from  the  action  of  the  wind  on  the  sail; 
if  we  consider  only  the  wind  components  parallel  and  perpendicular 
to  the  hull  we  are  dealing  merely  with  minor  influences, — one  acting 
on  the  stern  of  the  hull  pushing  the  ship  forward,  and  the  other  acting 
at  right  angles  to  the  hull  and  pushing  the  ship  out  of  her  course. 
Although  these  influences  have  little  to  do  with  propelling  the  ship, 
still  it  is  well  to  study  them  and  put  them  aside  before  considering 
the  action  of  the  wind  on  the  sails. 

Notice,  also,  that  the  question  asks  only  for  the  wind  component 
in  the  direction  of  the  ship.  This  cannot  be  found  from  the 
conditions  of  the  problem  unless  we  assume  something  concerning 
the  other  component.  The  solution  of  a  triangle  requires  that  at 
least  three  parts  be  given.  In  this  case  if  OA ,  Fig.  i  i-b,  represents  the 


Q         9 


Fig.  n-b 


wind  velocity  in  magnitude  and  direction,  and  a  horizontal  line,  OX, 
is  parallel  to  the  direction  of  the  ship,  we  have  as  given  conditions 
only  the  line  OA  and  the  angle  7.  The  magnitude  of  the  horizontal 
component  may  be  almost  anything,  —  depending  upon  what  conditions 
are  attached  to  the  other  component.  (See  pp.  15  and  16). 

When  not  otherwise  specified  it  is  customary  to  take  the  two 
components  at  right  angles  to  each  other.  In  this  case,  when  asked 
to  find  the  component  parallel  to  the  direction  of  the  ship,  we  have 


COMPOSITION   OF   VELOCITIES.  19 

assumed  that  the   other  component  is  at  right  angle  to  the  ship. 
(Fig.  n-c). 

(£)     What  are  the  wind  components  parallel  and  perpendicular 
to  the  sail  ?     (Fig.  12-0} 


Fig.  12-a 


Fig.  i2-b 


(c)  Now  the  component  a,  pushing  square  against  the  sail,  is 
not  entirely  useful  in  moving  the  ship  forward ;  on  the  contrary  the 
ship  tends  to  move  bodily  in  the  direction  of  component  a.  How  much 
is  component  a  acting  in  the  direction  of  the  ship,  and  how  much  is  it 
acting  perpendicular  to  that  direction  ?  (See  Fig.  I2-V). 


11.  The  Kite  Problem.  A  kite  flying  in  a  wind  blowing  22 
miles  per  hour,  is  inclined  at  an  angle  0/58°  with  the  horizon,  as  angle 
ft  in  Fig.  fj-a. 

(a)  Assuming  that  the  wind  is  blowing  horizontally,  what  are 
its  components — a  perpendicular,  and  b  parallel,  to  the  kite  f 


Fig.  i3-b 


(£)  The  component  perpendicular  to  the  kite  represents  the  total 
Pull  against  the  string  exerted  by  the  wind,  (assuming  that  the  string 
is  perpendicular  to  the  kite].  A  part  of  this  pull  is  upward  and  a 
part  horizo  ntal.  Find  each  part,  h  and  v,  Fig.  ij-b. 

This  part  v,  determined  in  this  manner,  is  the  lifting  compo- 
nent of  the  wind  on  the  kite. 


20  THEORETICAL   MECHANICS. 

12.  //  a  body  travels  d  miles  in  t  hours,  what  is  its  velocity  in 
feet  per  second? 

13.  If  a  body  has  a  velocity  of  v  feet  per  second,  in  how  many 
hours  will  it  travel  a  distance  of  m  miles  ? 

14.  Find  the  resultant  of  two  perpendicular  components,  one 
p  meters  per  minute,  and  the  other  q  feet  per  second. 


"Triangle  of  Velocities"  and  "Polygon  of  Velocities." 
These  two  ideas  find  frequent  expression  in  mechanics. 
The  former  has  been  referred  to  incidentally  in  a  previous 
paragraph,  and  is  now  reverted  to  for  the  purpose  of 
directing  attention  to  a  consideration  that  will  serve  to 
facilitate  the  extension  of  the  subject  of  composition  of 
velocities  to  cases  in  which  there  are  more  than  two 
components. 

As  already  stated,  the  resultant  of  two  components,  a 
and  b,  is  one  of  the  diagonals  r  of  a  parallelogram  OACB 
(Fig.  15).  In  problems  concerned  with  a,  b  and  r,  it  is 


Fig.  15- 


customary  to  solve  one  of  the  triangles,  OBC  or  OAC.  If 
we  work  from  the  triangle  OBC  we  do  not  deal  directly 
with  component  a,  but  with  BC,  parallel  and  equal  to  a;  or, 
if  we  choose  to  work  from  triangle  OA  C,  we  consider  AC  in 
place  of  component  b. 

Now,  what  is  signified  by  this  substitution  of  a  line  parallel 
and  equal  to  one  of  the  components  for  the  component  itself? 


COMPOSITION   OF   VELOCITIES. 


21 


It  has  already  been  emphasized  (p.  11)  that  the  very  idea  of 
the  composition  of  velocities  presupposes  that  the 
component  actions  take  place  simultaneously.  In  the  figure 
of  the  parallelogram  this  conception  of  simultaneous  action 
is  easily  comprehended  ;  but  it  is  abandoned  for  the  time 
being  as  soon  as  we  transfer  our  thoughts  to  one  of  the 
triangles  alone.  For  instance,  take  the  triangle  OA  C  apart 
from  the  rest  of  the  parallelogram.  The  resultant  is  found 

mathematically  from  OA  and 
AC,  Fig.  16,  as  if  resulting 
from  these  two  motions  taken 
consecutively,  whereas,  i  n 
truth,  the  component  motions 
J6.  are  not  only  not  consecutive, 

but  AC  is  not  one  of  them  at  all.  This  misconception 
grows  out  of  the  tendency,  already  referred  to,  to  assume 
that  OA  and  A  C  represent  distances  or  displacements,  instead 
of  velocities.  The  "triangle  of  velocities"  should  be  used 
with  a  clear  understanding  of  the  meaning  of  each  line. 
As  a  mathematical  expedient,  the  "triangle  of  velocities" 
is  a  simple  case  of  a  more  general  construction,  the 
polygon  of  velocities,  which  suffices  not  merely  for  two, 
but  for  any  number  of  components.  The  idea  is  simple, 
and  q  (Fig.  iy-a)  be  two  components  with  resultant  r. 


P 
Fig.  17- 


P 
Fig.   iy-b 


Now,  if  we  draw  a  line  OA  (Fig.  iy-b)  to  represent  p; 
from  A,  a  second  line  AC  parallel  and  equal  to  q;  and 
thence  from  C  a  line  CO  to  the  starting  point,  we  will  have 


22 


THEORETICAL   MECHANICS. 


constructed  a  closed  figure  of  which  the  closing  line  is 
parallel  and  equal  to  the  resultant  r,  but  drawn  in  the 
opposite  direction.  For  the  sake  of  convenience,  we 
temporarily  lay  aside  all  thought  of  simultaneous  action  and 
consider  the  components  successively,  and  we  thereby  reach 
a  conclusion  which  we  know  bears  a  certain  relation  to  the 
resultant, — a  line  equal  to  the  resultant  but  drawn  in  an 
opposite  direction. 

A  similar  device,  really  the  same  process,  can  be  used 
for  any  number  of  components.  I/et  p,  q,  s  and  /  (Fig.  18) 
be  four  components.  By  resorting  to  the  parallelogram 


Fig.  1 8- a 


Fig.  i8-b.    ' 


method  we  can  get  the  resultant  /  of/  and  q;  then  we  can  treat 
r'  as  a  single  motion  and  combine  it  with  s  in  the  same  way, 
getting  a  new  resultant  r"\  this  in  turn  can  be  combined 
with  t  to  obtain  the  final  resultant  r'". 

By  using  the  polygon  of  velocities  we  could  have 
determined  this  resultant  much  more  readily.  Draw  a  line 
equal  and  parallel  to  p  (Fig.  i8-b);  from  the  extremity  of 


COMPOSITION   OF  VELOCITIES.  23 

this  draw  a  second  line  equal  and  parallel  to  q;  thence  a  line 
equal  and  parallel  to  s;  and  one  equal  and  parallel  to  t. 
The  line  necessary  to  close  this  polygon  will  be  equal  to  the 
desired  resultant  but  opposite  in  direction. 

While  the  "polygon  of  velocities"  simplifies  the  graphical 
determination,  it  would  still  be  necessary  to  break  up  the 
figure  into  triangles  in  order  to  accomplish  a  trigonometric 
solution. 


Examples : 

1.  How  would  it  have  been  if  we  had  combined  the  components 
of  Fig.  i8-a  in  a  different  order?        Try  it,   assuming  any  four 
velocities  and  finding  their  resultant  by  combining  them  graphically 
in  at  least  two  different  orders  by  the  parallelogram  method. 

2.  If  the  lines  of  Fig.   iS-b   were   constructed  in    a  different 
order,  would  the  result  be  the  same  f 

3.  Find  the  resultant  motion  of  a  body  which  has  the  following 
component  motions  : 

8  miles  per  hour  from  W.  to  E.; 
3  miles  per  hour  34*  W.  of  S.; 
5.5  miles  per  hour  //°  S.  of  W. 

Construct  the  "polygon  of  velocities" 
to  accurate  scale,  and  compare  the 
graphical  result  with  the  trigonometric 
solution. 

4.  A  body  has  four  motions  measured 
in  meters  per  second,  as  indicated  by  the 
numbers  in  Fig.  19. 

Find  the  resultant  by  construction 
Fig.  19.  and  by  computation. 


24 


THEORETICAL  MECHANICS. 


Resultant  for  Angles  Greater  than  90°.  It  is  frequently 
asked,  "How  can  the  resultant  be  less  than  one  of  the 
components?"  This  is  often  the  case  when  the  angle 
between  two  components  is  greater  than  90°.  Let  the  two 
components  be  a  and  b  (Fig.  20-a).  Now,  for  convenience, 


Fig.  2o-a. 


6-c 
Fig.  2o-b. 


we  may  project  a  horizontally  and  vertically,  —  that  is,  regard 
it  as  the  resultant  of  two  components  c  and  d,  at  right 
angles  to  each  other.  From  this  it  is  evident  that  the 
component  a  is  acting  against  b  to  some  extent,  for  the 
reason  that  b,  c,  and  d  together  are  equivalent  to  b  and  a 
Subtracting  c  from  b,  we  have  left  a  horizontal  velocity 
represented  by  the  difference  between  c  and  b,  which, 
combined  with  d  (Fig.  20-b),  will  give  a  resultant  identical 
with  the  resultant  of  b  and  a. 

Algebraically,  this  is  shown  from  the  formula, 


cosO. 


If  0  is  greater  than  90°,  cos  0  is  negative.  Hence,  when 
2ab  cos  0  becomes  greater  than  a2,  then  the  resultant  r2 
becomes  less  than  b2;  and  when  2ab  cos  0  becomes  greater 
than  b2,  r2  becomes  less  than  a2. 


CHAPTER  III. 
CIRCULAR    MOTION. 

Motion  of  a  body  in  a  circle.  It  has  been  stated  (p.  3) 
that  a  full  description  of  the  motion  of  a  body  involves  : 

1 .  Its  direction  of  motion  ; 

2 .  Its  rate  of  motion . 

A  body  moving  in  a  circle  may  have  uniform  speed,  but 
its  motion  differs  from  the  rectilinear  motions  heretofore 
considered  by  constantly  changing  in  direction.  It  is  only 
necessary  to  recall  the  familiar  geometrical  distinctions 
between  straight  lines,  broken  lines  and  curved  lines,  in 
order  to  comprehend  the  difference  between  rectilinear  and 
curvilinear  motion, — the  former  constant  in  direction,  and 
the  latter  constantly  changing  direction.  Motion  in  a 
circle  at  a  uniform  speed  is  the  simplest  of  curvilinear 
motions,  because  the  change  of  direction  takes  place  at  a 
uniform  rate. 

The  speed  is  found  in  the  usual  way,  v  =  dlt.  For 
instance,  if  a  body  is  moving  in  the  circumference  of  a 
circle  with  radius  equal  to  r  feet,  and  requires  six  seconds 
for  completing  one  revolution,  its  velocity  is  2irr/6 
feet  per  second.  The  distance  traveled  per  second  is  ^  of 
the  circumference,  or  in  angular  measure  60°,  as  shown 
by  dotted  radius  r  (Fig.  21).  If  the  radius  had  been  twice 
as  large,  the  body,  moving  at  the  same  speed,  would 
traverse  an  arc  of  only  30°  in  one  second  ;  and  if  the  radius 
were  half  as  large,  the  angular  change  would  be  120°.  The 
rate  of  angular  change,  therefore,  depends  upon  the  radius 


26  THEORETICAL   MECHANICS. 

of  the  circle  as  well  as  upon  the  speed  of  the  body  ;  it  is 


Fig.  21. 


Fig.  22. 


directly  proportional  to  one  and  inversely  proportional  to 
the  other. 

Properly,  this  rate  of  angular  change  should  be 
determined  from  tangents  •  rather  than  from  the  radii,  as 
illustrated  in  Fig.  22.  When  the  body  moving  in  the  circle 
is  at  point  A,  its  motion  for  the  instant  is  in  the  direction  of 
the  tangent  A  T;  at  point  B  its  motion  is  in  the  direction  of 
the  tangent  BT.  If  the  arc  AB  is  60°,  the  angle  TCT'  is 
also  60°.  If  the  body  moves  in  the  circumference  from  A 
to  B  in  one  second,  it  not  only  has  a  velocity  of  2  tr  r/6 
feet  per  second,  but  the  direction  of  its  motion  is  also 
changing  at  the  rate  of  60°  per  second.  At  the  end  of  2 
seconds  the  body  will  have  covered  a  distance  AD  equal 
to  2ABy  and  the  total  change  of  direction  will  be  120°. 

The  rate  of  angular  change, 

total  change  of  direction 
time 

is  frequently  called  the  Angular  Velocity.     In  this  sense  the  word 


CIRCULAR    MOTION.  27 

velocity  does  not  have  its  usual  meaning  of  distance  traveled  per 
unit  of  time,  but  is  given  a  much  broader  significance,  equivalent  to 
the  general  expression  "rate  of  change."  This  is  a  figurative  use  of 
the  word  that  may  be  somewhat  confusing  to  beginners.  For 
example,  "temperature  velocity"  would  mean  the  rate  at  which  the 
temperature  is  changing.  In  the  same  way,  "angular  velocity" 
means  the  rate  at  which  the  direction,  (difference  of  direction,  or 
angle),  changes. 


Examples : 

1.  .If  the  earth's  radius  is  4,000  miles,  what  is  the  velocity  in 
miles  per  hour  of  a  point  on  the  equator,  due  to  the  earth1  s  axial 
rotation  ?     What  is  its  angular  velocity? 

2.  What  lineal  velocity  and  rate  of  angular  change  of  the  Lick 
Telescope  (situated  in  latitude  j7°  20'  25"  North],  is  caused  by  the 
ear th^s  axial  rotation? 

3.  The  moon  is  about  250,000  miles  from  the  earth,  and  completes 
its  revolution  in  about  28  days.       What  is  its  orbital  velocity  in  miles 
per  hour?     What  is  its  angular  velocity  ? 

4.  The  distance  from  the  earth  to  the  sun  is  02,000,000  miles. 
If  there  were  exactly  365  days  in  a  year,  what  would  be  the  orbital 
velocity  of  the  earth's  center  and  its  angular  velocity  ? 

5.  Two  gear  wheels  having  diameters  of  4  inches  and  12  inches, 
respectively,  have  fixed  axes  O  and  O'.     If  the  larger  wheel  has  200 
revolutions  per  minute,  how  many  revolutions  does  it  impart  to  the 
smaller  wheel?      What  is  the  lineal  velocity  of  each  point  on   the 
circumference  of  the  small  wheel,  and  what  is  its  angular  velocity 
about  Of     What  is  the  lineal  velocity  of  each  point  on  the  circumference 
of  the  larger  wheel,  and  what  is  its  angular  velocity  about  Q'?     Would 
the  re  stilts  have  been  the  same  if  the  motion  had  been  transmitted  from 
O'  to  0  by  belt  instead  of  by  gear  ? 

6.  A  lathe  is  connected  with  a  system  of  shafting,  as  illustrated 
in  Fig.   23,  the  numbers  indicating  the  diameters  of  the  pulleys  in 


28 


THEORETICAL   MECHANICS. 


inches.     The  speed  of  the  main  shaft  is  200  revolutions  per  minute, 
(a]     Find  the  revolutions  per  minute,  and  the  lineal  velocity 
of  the  perimeter  of  each  of  the  pulleys  mentioned  in  the  following 
table : 


PULLEY 


ANGULAR  VELOCITY  j     LINEAL  VELOCITY 
REVOLUTIONS 

PER  MINUTE    |  FEET  PER    MINUTE 


18  inch 

24  " 

9  " 

10  " 

5i  " 

64  " 

8  " 

94  " 


(b]     Find  the  angular  velocity  of  chuck^  with  and  without 
back-gear.     Tabulate  results  as  follows : 


POSITION  OF 

Without  Back  Gear 

With  Back  Gear 

BELT 

Angular  Velocity 
Cone  and  Chuck 

Angular  Velocity 
of  Back  Gear 

Angular  Velocity 
of  Chuck 

On  7^r  inch  Pulley 

"     6       " 

11    4i     l- 

• 

"     3       " 

CIRCULAR    MOTION. 


AUXILIARY    SHAFT 


-»    COUNTERSHAFT 


LATHE 


Teeth 


18 


MAIN  SHAFT        sooRev. 
Fig.  23. 


Min. 


Composition  of  Circular  and  Rectilinear  Motions.  When 
a  ball  is  thrown  in  any  direction  it  usually  possesses 
a  rotary  or  whirling  motion,  in  addition  to  and  in- 
dependently of  its  progressive  motion  or  motion  of 
translation.  It  is  by  his  ability  to  produce  and  control  these 
rotations  that  a  skillful  baseball  pitcher  causes  a  baseball  to 
move  in  an  erratic  path,  —  "pitching  curves,"  as  it  is  called. 
If  the  ball  is  moving  in  a  straight  line  and  rotating  around 
an  imaginary  axis  passing  through  its  center,  the  actual 
motion,  relatively  to  the  ground,  of  any  point  on  its  surface 
at  any  instant  is  the  resultant  of  a  circular  or  rotational 
motion  and  a  rectilinear  motion  or  translation.  A  simple 
case  is  assumed  in  the  example  on  the  following  page. 


30  THEORETICAL   MECHANICS. 

Example : 

The  Baseball  Problem.     A  baseball,  2.75  inches  in  diameter, 

is  thrown  horizontally  with  a 
velocity  of  80  feet  per  second, 
and  at  the  same  time  is  made  to 
rotate  at  the  rate  of  jo  revolu- 
tions per  second  around  a  hor- 
izontal axis  at  right  angles  to 
the  direction  in  which  it  is 
thrown. 

In    Fig.  23,   let    the    circle 

Fig.  24.  ABCD  represent  a  vertical  section 

through  the  center  of  the  ball. 
The  line  OX  represents  the  horizontal  velocity,  and  the  curved  arrow 
indicates  a  rotation  around  an  axis  imagined  to  pass  through  O, 
perpendicular  to  the  plane  of  the  paper. 

(a)  Find  the  resulant  velocity  of  each  of  the  points  A,  B,  C 
and  D,  at  the  instant  when  they  are  in  the  position  shown  in  the 
figure. 

DISCUSSION: — The  entire  mass  of  the  ball,  including  the  points 
A,  B,  C  and  D,  moves  horizontally  with  the  same  velocity,  OX,  as 
the  center,  and  hence  OX  is  one  of  the  components  for  each  of  these 
points. 

The  other  component,  due  to  the  rotation  of  the  ball,  is  different 
for  each  point  of  the  ball ;  for  all  points  on  the  circumference  of  the 
circle  ABCD,  the  rate  or  magnitude  of  this  rotational  component  is 
the  same,  but  no  two  points  are  moving  in  the  same  direction  at  the 
given  instant.  When  C  is  at  the  lowest  position  on  the  baH,  the 
rotation  gives  it  a  motion  for  the  instant  in  the  direction  of  the 
tangential  arrow  CC'\  the  rotational  component  of  point  D  at  that 
instant  would  be  vertically  upward  ;  of  A  horizontally  to  the  right ; 
and  of  B  vertically  downward. 

It  is  important  that  the  student  should  get  a  clear  conception  of 
what  is  meant  by  these  tangential  components.  True,  the  points 
move  in  the  circle  and  not  along  the  tangential  lines.  They  follow 
the  circular  path,  however,  merely  because  they  are  constrained  to  do 
so.  It  is  not  difficult  to  see  that,  as  the  ball  revolves  and  C  moves 


CIRCULAR    MOTION.  31 

from  its  present  position  to  that  now  occupied  by  D,  if  at  the  instant 
when  it  reaches  the  latter  point  it  is  freed  from  this  constraint  (which 
would  pull  it  around  in  the  circle  towards  A],  it  will  move  instead 
along  the  tangent  DD', — like  a  drop  of  water  similarly  freed  and  flying 
from  a  grindstone. 

Now,  while  the  points  A,  B,  C  and  D  are  not  thus  freed  from  the 
ball,  still  their  respective  motions  are  for  the  instant  in  the  directions 
of  the  tangents.  Imagine  the  circle  made  up  of  an  infinite  number 
of  very  small  straight  lines — a  regular  polygon  of  an  infinite  number 
of  sides — and  the  same  conclusion  is  reached.  The  infinitesimal  line 
constituting  any  part  of  the  circumference  will  have  the  direction  of 
the  tangent  at  that  point.  The  rotational  component  of  D  is  vertically 
upward  for  only  an  infinitesimal  fraction  of  a  second,  and  then  it 
assumes  a  different  direction.  But,  if  we  wish  to  combine  this 
rotational  component  with  the  horizontal  component  OX,  we  must 
represent  it  graphically  on  the  same  scale.  Hence,  if  OX  represents 
one  of  the  velocities  of  D  in  feet  per  second,  then  the  other,  in 
direction  DD't  must  also  be  represented  in  feet  per  second.  The 
point  D  does  not  continue  to  move  in  direction  DD'  for  a  full  second, 
but  if  it  did  it  would  cover  the  distance  DD',  on  the  same  scale  as  OX. 
When  we  say  that  a  train  has  a  velocity  of  30  miles  per  hour,  we  do 
not  insist  that  it  shall  move  the  full  hour,  but  we  can  represent  its 
motion  graphically  as  if  it  did ;  so  we  say  of  the  ball,  if  point  D 
moves  a  very  small  distance  in  direction  DD'  in  a  very  small  interval 
of  time,  at  the  same  rate  it  would  in  a  full  second  move  the  distance 
DD'. 

Hence,  the  motion  of  D,  relatively  to  the  ground  at  the  given 
instant,  is  the  resultant  of  the  two  components  OX  and  DD  at  right 
angles  to  each  other.  And  this  will  determine  the  direction  of  the 
resultant  as  well  as  its  magnitude.  The  two  components  for  B  are 
also  at  right  angles  to  each  other ;  those  for  A  and  C  are  parallel. 

Notice  that  the  ball  is  represented  in  Fig.  24  on  a  scale  much 
larger  than  the  scale  of  velocities.  If  it  makes  30  rotations  per  second 
it  thereby  causes  the  points  A,  B,  C,  and  D  to  move  in  a  second 
through  a  distance  equal  to  30  times  the  circumference  shown  in  the 
figure, — which,  as  it  is  represented  in  the  diagram,  is  many  limes 
greater  than  the  line  DD'.  It  is  obvious  that  the  velocities  had  to  be 
represented  on  a  greatly  reduced  scale. 

($)     Find   the    magnitude  and   direction   of  the    resultant 
'velocity  of  a  point  on  the  ball  between  A  and  B,  22°  from  B. 


32  THEORETICAL   MECHANICS. 

The  motion  of  a  carriage  wheel  is  not  unlike  that  of  the 
ball  in  the  preceding  example,  except  that  the  rate  of 
rotation  of  the  wheel  is  not  independent  of  the  velocity 
of  the  vehicle.  A  baseball  may  revolve  fast  or 
slowly,  without  regard  to  the  velocity  with  which  it  is 
thrown,  but  the  rate  at  which  a  carriage  wheel  revolves 
necessarily  bears  a  fixed  relation  to  the  velocity  of  the 
vehicle.  In  one  revolution  the  wheel  measures  the  length 
of  its  circumference  on  the  ground.  Hence,  the  rotational 
component  of  any  point  on  the  perimeter  of  the  wheel  is 
exactly  equal  to  the  rectilinear  component  parallel  to  the 
ground.  Otherwise,  the  general  conditions  are  the  same  as 
those  assumed  for  the  baseball  in  the  preceding  example. 

/ 
Examples : 

1.  We  know  that  that  point  on  the  tire  of  a  wheel  momentarily 
touching  the  ground  is  at  rest  relatively  to  the  ground*;  otherwise 
there  would  be  slipping.     Prove    this    by    the    composition    of  the 
rectilinear  and  rotational  components. 

2.  If  v  is  the  velocity  of  the  vehicle,  prove  that  the  highest  point 
on  the  wheel  has  a  resultant  velocity  of  2v  in  the  same  direction. 

3.  A  carriage  is  traveling  at  the  rate  of  g  miles  per  hour. 

(a]  Find  the  resultant  velocity  relatively  to  the  ground  of  a 
point  on  the  front  side  of  one  of  the  wheels  46"  from  the  ground.. 

(b)  If  the  diameter  of  the  front  wheels  is  3%  feet  and  of  the 
hind  wheels  4  feet,  find  the  angular  velocity  of  each. 

Composition  of  two  Circular  Motions.  Every  point  on 
the  earth's  surface  is  subject  to  at  least  two  motions,  one 
due  to  the  rotation  of  the  earth  on  its  axis,  and  the  other 
due  to  its  orbital  motion  or  revolution  around  the  sun. 


*  If  there  is  any  doubt  of  this  in  the  mind  of  the  student,  try  the  experiment 
of  tying  a  piece  of  white  cloth,  or  marking  with  a  piece  of  chalk,  around  a  bicycle 
tire.  It  will  be  seen  that  when  the  bicycle  is  ridden,  the  cloth  or  chalk  mark  comes 
to  rest  as  it  reaches  the  ground. 


CIRCULAR    MOTION. 


33 


Hence,    the   resultant   motion  of  a   point   A    (Fig.    25)    is 
different  from  that  of  a  point  B. 


Fig-  25. 

As  the  moon  revolves  around  the  earth,  however,  it  does 
not  possess  an  additional  independent  motion  around  its  own 
axis.  Its  motion  in  this  respect  is  analogous  to  the  motion 
of  a  circular  disc  fastened  to  a  stick,  as  in  Fig.  26.  If  this 
whole  device  is  made  to  revolve  around  point  O  of  the  stick 
in  the  direction  of  the  arrow,  a  person  standing  beyond  A 
would  see  successively  points  A,  £,  C,  D.  To  him, 
therefore,  there  would  be  an  apparent 
rotation  of  the  disc  around  its  center, 
but  this  apparent  rotation  is  really 
controlled  by  the  revolution  about  O-  ^ 

This  does  not  include  the  considera- 
tion that  the  moon  is  carried  with  the 
earth  in  the  motion  of  the  latter  around 
the  sun,  and  hence  has  a  double 
motion  after  all.  In  fact,  if  the 
moon  had  an  independent  axial 
rotation  each  point  in  it  would  then 
be  subject  to  three  simultaneous, 
circular  motions. 

Since  the  earth's  axis  is  inclined 
to  its  orbit,  and  since  the  moon's 
orbit  around  the  earth  is  inclined  to 
the  earth's  orbit,  the  determination  of 
the  resultants  of  these  motions  is  not  Fig.  26. 


34 


THEORETICAL  MECHANICS. 


simple  enough  for  our  purposes  of  explanation.  The  same 
ideas,  however,  are  illustrated  in  modified  form  in  the 
following 


Examples : 

1.     A  circular  disc  6  inches  in  diameter  is  fastened  to  a   stick. 
This  device  is  made  to  revolve  25  times  per  minute  around  an  axis 
which  passes  through  the  stick  at  a  point  18 
inches  from  the  center  of  the  disc.     Find  the 
tangential  velocity  of  the  center  of  the  disc 
and  of  each  of  four  points  situated  as  A,  B, 
^  C  andT)y  Fig.  26,(line  BD  being  perpendicular 
to  AC.) 

2.  Now  imagine  the  screws  removed 
and  the  disc  no  longer  fixed  rigidly  to  the 
stick,  but  free  to  rotate  aroimd  a  pin  P 
at  its  center,  as  in  Fig.  27.  If  the  disc 
rotates  70  times  per  minute  around  P,  in 
addition  to  its  revolution  arotmd  O,  what 
will  be  the  resultant  velocity  of  each  of  the 
points  P,  A,  B,  and  C? 

This  is  an  arrangement  that  would  be 
quite  analogous  to  the  axial  and  orbital 
motions  of  the  earth,  if  the  earth's  axis 
were  perpendicular  to  the  plane  of  its  orbit. 
Comparing  this  example  with  the  base- 
ball problem  will  show  how  the  composition 
of  two  circular  motions  differs  from  the 
composition  of  a  circular  with  a  rectilinear  motion.  The  revolution 
of  the  disc  (Fig.  27)  in  a  circular  path  around  O  gives  point  A  a 
greater  component  than  point  C,  but  if  the  disc  were  moving  in  a 
straight  line,  like  the  baseball  in  Fig.  27,  this  rectilinear  component 
would  be  the  same  for  all  points. 

In  the  baseball  problem  every  point  on  the  perimeter  of  the 
section  shown  in  the  figure  had  the  same  pair  of  components,  the 
only  difference  for  different  points  being  in  the  angle  between  the 
two  components;  in  Fig.  27,  not  only  is  the  angle  different  for  each 


Fig.    27. 


CIRCULAR    MOTION. 


35 


pair  of  components,  but  only  one  of  the  components  is  the  same  for 
all  points. 


3.  If  the  earth' s  axis  were  perpendicular  to  the  plane  of  its 
orbit,  what  would  be  the  resultant  velocity  of  a  point  on  the  equator 
at  midnight,  at  noonday,  and  at  7  o 'clock  p.  m.  ? 

Assume  :  One  year  equal  to  365  days ;  diameter  of  the  earth  at 
the  equator  equal  to  8,000  miles ;  distance  of  the  center  of  the 
earth  from  the  sun  equal  to  92,000,000  miles  ;  that  the  earth  revolves 
around  the  sun  from  west  to  east  and  has  its  axial  rotation  in  the 
same  direction. 


4.  If  the  moot? s  orbit  around  the  earth  were  in  the  same  plane 
as  the  earths  orbit,  what  would  be  the  resultant  velocity  of  the  center 
of  the  moon  at  "new  moon"? 


Assume  that  the  moon  completes  its  circuit  around  the  earth  in 
28  days,  and  that  the  motion  is  from  west  to  east,  like  the  orbital 
motion  of  the  earth. 


28. 


5.  A  circle  6  inches  in  diameter  rolls 
around  a  larger  circle,  diameter  24.  inches, 
at  the  rate  of  50  revolutions  per  minute. 
B  is  the  center  of  the  small  circle,  and  A  and 
C  points  on  Us  circumference.  What  are  the 
lineal  and  angular  velocities  of*&?  What 
is  the  resultant  lineal  velocity  of  K  and  of  'Cf 

This  problem  bears  the  same  relation 
to  the  three  examples  preceding  that  the 
carriage  wheel  motion  bears  to  the  baseball 
problem  ;  the  axial  rotation  of  the  earth  is 
not  in  a  simple  ratio  to  its  orbital  motion, 
and  is  independent  of  it,  but  in  this  problem 


the  two  motions  of  the  small  circle  are  inter-dependent,  if  there  is  no 
slipping. 


36  THEORETICAL   MECHANICS. 

Cycloids.  If  a  piece  of  chalk  or  other  marker  be 
fastened  at  the  perimeter  of  a  disc,  and  the  latter  be  rolled 
along  the  floor  with  the  disc  against  the  wall,  the  marker 
will  trace  a  line  like  the  curve  shown  in  Fig.  29.  This 
curve  is  called  a  Cycloid. 


Fig.  29. 

Every  point  on  a  carriage  wheel  describes  such  a  path. 
An  inspection  of  this  curve  discloses  the  fact,  already 
referred  to,  that  every  point  on  the  perimeter  of  the  disc  or 
wheel  comes  to  rest  as  it  reaches  the  ground.  Approaching 
the  ground,  it  is  moving  almost  vertically  downward,  and 
as  soon  as  it  begins  to  leave  the  ground  its  motion  is 
upward.  Having  turned  back  abruptly,  as  the  diagram 
shows,  it  must  have  come  to  rest  at  the  turning  point. 


GENERATING  CIRCLE 


Fig.  30. 


CIRCULAR    MOTION. 


37 


The  cycloid  generated  by  a  point  on  the  circumference 
of  a  circle  as  it  rolls  along  a  straight  line  is  only  one  of  a 
class  of  similar  curves  that  are  used  in  mechanics,  especially 
in  designing  gear  teeth.  If  a  circle,  instead  of  rolling  along 
a  straight  line,  be  rolled  on  the  circumference  of  another 
circle,  as  in  Fig.  30,  it  will  generate  a  curve  called  an 
Epicycloid.  If  it  be  rolled  on  the  inside  of  a  ring  in  the 
same  direction  as  before,  its  rotations  around  its  own  axis 
will  be  in  the  opposite  direction,  and  the  curve  generated  in 
this  case  is  called  a  Hypoeyeloid. 


GENERATING  CIRCLE 


FUNDAMENTAL  CIRCLE 


HYPOCYCLOID 


Fig.  31- 


For  methods  of  constructing  these  curves  geometrically  see 
Kent's  Mechanical  Engineers'  Pocket  Book,  pp.  49  and  50,  or 
Appleton's  Cyclopaedia  of  Drawing  and  Design,  pp.  303  and  304. 


CHAPTER  IV. 
ACCELERATION. 

A  perfectly  uniform  rate  of  motion  is  seldom  realized. 
The  movements  of  a  clock  or  a  line  of  shafting  are  generally 
assumed  to  be  uniform,  but  are  really  subject  to  many 
fluctuations.  An  electric  car  moves  with  increasing  velocity 
up  to  the  desired  speed,  maintains  an  apparently  uniform 
velocity  for  a  time,  and  again  assumes  a  distinctly  variable 
velocity  before  stopping.  A  falling  body  gains  velocity 
continuously  ;  and  if  hurled  upward,  loses  velocity  steadily 
until  it  again  starts  downward. 

The  subject  of  acceleration  deals  with  all  cases  of 
variable  velocity.  When  the  rate  of  motion  changes 
irregularly  or  spasmodically  it  generally  involves  too  many 
complexities  for  treatment  by  arithmetical  or  algebraic 
methods.  Hence,  our  study  of  acceleration  will  be  limited 
to  motions  that  change  uniformly, — called  uniformly 
accelerated  motions. 

It  should  be  remembered  that  changes  of  direction,  as  exemplified 
by  motion  in  a  circle,  need  not  affect  the  rate  of  a  motion.  Unless 
otherwise  specified,  a  motion  is  alwaj'S  assumed  to  be  in  a  straight 
line. 

Suppose  that  a  body 

4   ^  B  is  observed  at  a  given 

instant  to  have  a  veloc- 

^  y  Q     ity    of    10    miles    per 

hour,    represented  by 
Fig'  32'  the  line  AB,  Fig.  32. 


ACCELERATION. 


39 


If,  after  a  few  minutes — say  three  minutes,  it  is  found 
to  have  a  velocity  of  16  miles  per  hour,  represented  by  the 
line  CD,  the  question  arises:  Was  this  additional  velocity 
caused  by  some  sudden  impulse,  or  was  it  a  steady  increase 
distributed  equally  over  the  entire  interval  of  three  minutes  ? 
If  we  take  the  latter  supposition,  and  divide  the  total 
increase  of  velocity  (6  miles  per  hour)  by  the  time  during 
which  the  increase  took  place  (3  minutes),  the  quotient  will 
represent  the  amount  of  velocity  (in  miles  per  hour)  by 
which  the  motion  of  the  body  was  increased  during  each  of 
the  three  minutes,  or  its  acceleration  in  miles  per  hour,  per 
minute.  At  the  end  of  the  first  minute  the  velocity  must 
have  been  1 2  miles  per  hour  ;  at  the  end  of  the  second 
minute,  14  miles  per  hour  ;  etc.  At  the  end  of  i^  minutes 
it  was  13  miles  per  hour  ;  at  the  end  of  2^  minutes  it  was 
14.5  miles  per  hour  ;  etc.  The  rate  of  motion  is  not  the 
same  at  any  two  successive  instants,  and  the  body  does  not 
travel  through  any  appreciable  distance  at  a  fixed  rate. 

Some  of  these  things  can  be  shown  graphically.  Draw 
a  horizontal  base-line  OX  and  a  vertical  base-line  O  Y.  Let 
intervals  of  time  be  represented  on  a  convenient  horizontal 


Fig.  33- 

scale,  say  i  inch  =.  i  minute,  and  use  a  vertical  arrow  to 
indicate  the  velocity  at  any  instant,  on  a  scale  of  i  inch  = 
velocity  of  10  miles  per  hour.  (The  dimensions  of  Fig.  33 


40  THEORETICAL  MECHANICS. 

are  reduced  one-half.)  Oy  will  represent  the  velocity  as 
first  observed,  and  Oy.A  will  represent  the  velocity  as 
observed  after  the  lapse  of  3  minutes.  Ox3  will  represent 
the  lapse  of  3  minutes.  The  line  x3C,  parallel  to  OY, 
contains  all  points  3  inches  from  the  K-axis  ;  the  line  y.A  C, 
parallel  to  OX,  contains  all  points  1.6  inches  from  the 
^f-axis.  The  intersection  of  x.A  C  and  j/3  C  locates  a  point 
which  pictures  the  two  things,  viz., — that  at  the  end  of  3 
minutes,  Oxz,  the  velocity-  of  the  body  is  16  miles  per 
hour,  x3C.  To  show  that  the  rate  of  motion  has  increased 
uniformly  we  have  only  to  connect  y  and  C  by  a  straight 
line.  The  upward  inclination  of  yC  shows  that  the  rate  of 
motion  is  increasing,  and  the  steepness  of  the  gradient 
pictures  the  rapidity  of  the  change.  The  total  increase  of 
velocity  for  the  3  minutes  is  cC. 

Supposing  that  the  rate  of  motion  of  this  body  continues 
to  increase  at  the  same  rate  as  observed  for  the  interval  of 
3  minutes,  can  we  calculate  the  velocity  it  would  have  at 
any  subsequent  time  ?  If  the  rate  of  increase — 2  miles 
per  hour,  per  minute — continues,  then  the  velocity  at 
the  end  of  the  fourth  minute  from  the  beginning  will  .be 
10+  4  X  2,  or  18  miles  per  hour,  which  would  be  repre- 
sented by  the  line  x±D  in  the  diagram. 


Examples  : 

1.  Taking  a  point  xa  (Fig.  jj)  so  thatO-x.l  will  represent  one 
minute,  what  will  be  represented  by  the  perpendicular  xx  A  ?  What 
will  a  A  represent?  How  could  we  find  from  the  diagram  the  velocity 
of  the  body  at  the  end  of  7.7  minutes  ?  At  what  time  after  starting 
would  the  'velocity  be  /j.j  miles  per  hour  ?  What  does  dD  represent? 


2.  A  body  starting  from  rest  has  its  velocity  increased  gradually 
and  uniformly  until  at  the  end  cf  one  second  its  velocity  is  found  to  be 
10  feet  per  second. 


ACCELERATION.  41 

(a)  What  has  been  its  average  velocity  meanwhile  ? 

(b]  Suppose  its  velocity  to  increase  always  at  the  same  rate ; 
what  would  be  its  velocity  at  the  end  of  the  second  second? 

(c)  What  will  have  been  its  average  velocity  during  the  first 
two  seconds  ? 

(d]  What  average  velocity  during  second  second? 

(  e )     What  average  velocity  during  first  three  seconds  f 

(f)  What  average  velocity  during  third  second  ? 

3.  (a]     What  distance  will  be  passed  over  by  this  body  during 
first  second? 

(&)  During  the  first  two  seconds  ? 

( c]  During  the  second  second  ? 

(d]  IJliring  the  first  three  seconds  ? 
( e}  During  the  third  second  ? 

4.  A  train  of  cars  leaving  a  station  acquires  velocity  gradually 
and  uniformly  until  at  the  end  of  three  minutes  it  is  moving  at  the 
rate  of  jo  miles  per  hour. 

(a)  What  velocity  in  miles  per  hour  has  been  added  each 
minute  ? 

(I)}  What  velocity  in  feet  per  minute  has  been  added  each 
minute  ? 

(c]  What  velocity  in  feet  per  minute  has  been  added  each 
second? 

(d}  What  velocity  in  feet  per  second  has  been  added  each 
second? 

5.  A  body  starting  from  rest  is  accelerated  uniformly.    At  the 
end  of  5  seconds  it  has  a  velocity  of  100  feet  per  second. 

(a]  What  has  been   its  average  velocity    during   the  five 
seconds  ? 

(b]  How  far  has  it  traveled? 

(c]  What  was  its  velocity  at  the  end  of  the  first  second? 

(d]  What  is  its  acceleration? 

(e}  Now  far  did  it  travel  during  the  first  second? 

(f}  What  was  its  velocity  at  the  end  of  the  second  second  f 

(g)  How  far  did  it  travel  during  first  two  seconds  ? 
(h )  How  far  did  it  travel  during  second  second  ? 

(/)     How  far  did  it  travel  during  the  first  ten  seconds  ? 
(/)    How  far  did  it  travel  during  the  fourteenth  second  ? 


42  THEORETICAL   MECHANICS. 

Definition  of  Acceleration.  By  the  acceleration  of  a 
body  we  mean  the  rate  at  which  its  rate  of  motion,  or 
velocity,  changes. 

In  the  first  illustration  of  this  idea  at  the  beginning  of 
this  chapter,  the  velocity  changed  from  10  miles  per  hour  to 
1 6  miles  per  hour  in  3  minutes, — the  change  taking  place  at 
the  rate  of  2  miles  per  hour  for  each  minute  ;  and  hence, 
according  to  the  definition,  the  acceleration  was  2  miles  per 
hour,  per  minute. 

In  example  2,  p.  40,  the  acceleration  was  10  feet 
per  second,  per  second  ;  in  exercise  4  it  was  10  miles  per 
hour,  per  minute,  or  y£>  mile  per  hour,  per  second,  (and 
might  have  been  expressed  in  any  other  units  of  time  and 
distance,  as  feet  per  minute,  per  minute  ;  feet  per  minute, 
per  second  ;  feet  per  second,  per  second  ;  inches  per  second, 
per  second,  etc.) 

Formulae  for  Uniformly  Accelerated  Motion.  From  the 
preceding  examples  and  statements  we  can  generalize  to  the 
following  relations  : 

(i)  If  a  body  starting  from  rest  is  subject  to  a  uniform 
acceleration,  a,  its  velocity,  v,  at  the  end  of  any  time, 
t,  will  be 

V   =  at.  (4) 

(ii)  If  the  body  does  not  start  from  a  condition  of  rest 
its  motion  may  be  either  accelerated  or  retarded,  the  latter 
being  regarded  as  a  negative  acceleration.  If  its  initial 
velocity  is  vlt  the  velocity  at  the  end  of  any  given  interval, 
/,  will  be 

V   =Vl  -fa/.  (5) 

(iii)  To  find  the  distance  traveled  in  any  time  by  any 
body,  no  matter  how  it  is  moving,  we  multiply  the  average 


ACCELERATION.  43 

velocity  by  the  time.  If  the  body  starts  from  rest  and  is 
subject  to  a  uniform  acceleration  a,  the  velocity  at  the  end 
of  time  t,  is  at;  the  average  velocity  during  t  is  a//2. 
The  distance  traveled  is 


If,  instead  of  starting  from  rest,  the  motion  changes 
uniformly  from  an  initial  velocity  z^,  to  a  final  velocity  vz, 
then  the  distance  traveled  meanwhile  will  be 


But     v,,     can    be    expressed     in     terms    of    v1     and    the 
acceleration,  thus  : 

V2   =  Vl  -|-  at. 

Whence 


Examples  : 

1.  A  body  starting  from  rest  moves  with  a  uniformly  accelerated 
motion.     When  it  has  passed  over  a  distance  of  100  feet  its  velocity  is 

found  to  be  50  feet  per  second. 

(a]  What  has  been  its  average  velocity  meanwhile  ? 

(b]  How  long  has  it  taken  the  body  to  travel  the  100  feet  ? 

(c]  What  is  its  acceleration  ? 

(rtf)     What  was  its  velocity  at  the  end  0/2.6  seconds  ? 
(<?)    What  will  be  its  velocity  at  the  end  of  I  minute,  and  how 
far  will  it  have  traveled  meanwhile  ? 

2.  A    body  starting  with  a  velocity  of  100  feet  per  second  loses 
its  velocity  gradttally  and  uniformly  and  comes  to  rest  in  6  seconds. 

(a)  What  is  its  acceleration  ? 

(b)  How  far  will  it  have  traveled  before  coming  to  rest  ? 

(c)  How  far  will  it  have  traveled  at  the  end  of  3  seconds  ? 


44  THEORETICAL  MECHANICS. 

3.  A    body  starting  with   a    velocity  of  jo  feet  per  second  is 
accelerated  uniformly  at  the  rate  of  8  feet  per  second,  per  second. 

(a)     What  will  be  its  velocity  at  the  end  of  5  seconds  ? 

(b}     How  far  will  it  have  traveled  meanwhile  ? 

(c)     How  far  will  it  travel  during  the  thirteenth  second? 

4.  A  body  starting  with    a   velocity    of  jo   feet  per  second  is 
accelerated  uniformly.     At  the  end  of  j  seconds  it  has  a  velocity  of  200 
feet  per  second. 

(a]     What  is  its  acceleration  ? 

(b}     How  far  does  it  travel  during  the  second,    third  and 
fourth  seconds  inclusive  ? 

5.  A  body  starting  with  a   velocity  of  jo  feet  per  second  is 
accelerated  uniformly .      At  the  end  of  10  seconds  it  is  found  to  have 
traveled  2,000  feet. 

(a]     What  has  been  its  average  velocity  ? 
(b}     What  was  its  final  velocity  ? 
(c}      What  was  its  acceleration  ? 

6.  A  body  starting  with  a  velocity  of  50  feet  per  second  loses 
velocity  at  a  uniform  rate.     At  the  end  of  j  seconds,  it  has  a  velocity 
of  20  feet  per  second. 

(a]     What  was  its  acceleration  ? 

(b}     How  long  before  it  will  come  to  rest  ? 

(c]     How  far  will  it  move  before  coming  to  rest  ? 

7.  If  a  body  starts  from  rest  and  moves  through  the  distance  d 
while  it  is  acquiring  a  velocity  v,  prove  that 

d=~>  (8) 

where  a  is  the  acceleration. 

Acceleration  of  Gravity.  The  acceleration  of  gravity  is 
uniform  and  equal  to  about  32.2  feet  per  second,  per  second. 
It  is  usually  represented  by  the  letter  g.  Under  the 
influence  of  the  attraction  that  exists  between  the  general 
mass  of  the  earth  and  objects  on  its  surface,  a  falling  body 


ACCELERATION.  45 

gains  velocity  at  the  rate  specified  ;  a  body  hurled  upward 
loses  velocity  at  the  same  rate. 

The  value  of  g  is  different  for  different  parts  of  the 
earth's  surface.  It  is  greater  at  the  poles  than  at  the 
equator  and  is  subject  to  other  irregularities. 

Examples : 

1.  A  body  falling  from  a  certain  height  requires  j  seconds  to 
reach  the  grotmd.      With  what  velocity  does  it  strike  ?     What  was  its 
velocity  at  the  end  of  1.5  seconds  ?    At  the  end  o/J./'S  seconds  f    From 
what  height  was  it  dropped  ?     What  part  of  the  distance  was  covered 
in  each  of  the  five  seconds  ?       What  part  was  covered  in  the  first  7.7 
seconds  ?    In  the  first  4.4.  seconds  ?    During  the  interval  between  the  end 
of  the  second  second  and  the  end  of  j.8  seconds  ? 

2.  A  falling  body  strikes  the  ground  with  a  velocity  of  193.2  feet 
per  second.     From  what  height  was  it  dropped  f 

3.  A   body  is  dropped  from  a  height  of  jSS.q  feet.     How  long 
before  it  will  strike  the  ground  ?     With  what  velocity  will  it  strike  ? 

4>.  A  body  is  hurled  downward  with  a  velocity  of  100  feet  per 
second.  What  will  be  its  velocity  at  the  end  of  3  seconds  ?  How  far 
will  it  have  traveled?  How  far  will  it  have  traveled  at  the  end  of 
5  seconds  f  What  Part  of  this  distance  will  have  been  covered  during 
each  of  the  5  seconds  ?  Ditring  second  and  third  seconds,  inclusive  f 

5.  A  body  being  hurled  vertically  downward  strikes  the  ground 
at  the  end  of  7  seconds  with  a  velocity  of  525.4.  feet  per  second.     With 
what  velocity  was  it  hurled?    From  what  height  was  it  hurled? 

6.  A    body  is  hurled  downward  with  a  velocity  of  20  feet  per 
second  from  a  height  of  1810  feet.     How  long  before  it  will  strike  the 
ground? 

7.  A  body  is  projected  vertically  upward  with  a  velocity  of  1000 
feet  per  second.     How  long  will  it  continue  to  rise?    How  far  will  it 

rise  ?     With  what  velocity  will  it  strike  the  ground  on  returning? 


46  THEORETICAL  MECHANICS. 

8.  A   man  reaching  from   the  top  of  a  tower  644  feet  high, 
throws  a  ball  vertically  upward  with  a  velocity  of  96.6  feet  per  second. 
How  long  before  it  will  reach  the  ground  f 

9.  If  the  acceleration  of  gravity  is  g  feet  per  second,  per  second, 
express  the  following  relations : 

(  i  )  v  in  terms  of  g  and  t ; 
( ii )  h  in  terms  of  g  and  t ; 
(///')  h  in  terms  of  g  and  v. 

The  body  is  assumed  to  start  from  rest ;  v  is  its  velocity  at  the  end 
of  t  seconds,  and  h  is  the  distance  it  falls  meanwhile. 

10.  What  is  the  value  of  g  expressed  in  cm.  per  second,  per 
second^ 


Use  Of  Co-ordinate  Axes.  The  method  of  locating  points 
with  reference  to  two  lines  or  axes,  employed  in  Fig.  33,  p.  39,  and 
elsewhere  in  this  book,  is  used  very  widely  in  various  branches  of 
mathematics  and  science.  The  two  reference  lines  may  be  drawn  at 
any  angle  with  each  other,  although  they  are  usually  perpendicular, 
one  being  drawn  horizontally  and  the  other  vertical!}^.  Their 
intersection  is  called  the  origin.  The  horizontal  line  is  called  the 
X-axis,  or  axis  of  abscissas,  and  the  vertical  line  is  the  Y-axis,  or 
axis  of  ordinates. 

The  familiar  geographical  method  of  locating  places  by  means  of 
latitude  and  longitude  assumes  two  such  reference  lines — the 
terrestrial  equator  and  the  arbitrary  line  through  Greenwich  called 
the  zero  meridian.  What  do  we  mean,  for  example,  when  we  say 
that  a  place  is  situated  in  latitude  30°  N.,  longitude  80°  W.?  The 
thirtieth  parallel  of  north  latitude  is  the  location  or  locus  of  all 
points  30°  north  of  the  equator,  and  the  eightieth  meridian  west  is 
the  locus  of  all  points  80°  west  of  the  zero  or  Greenwich  meridian. 
The  intersection  of  these  two  lines  or  loci  is  the  point  described. 

Tracts  of  public  land  are  located  in  the  same  manner.  In  the 
central  portions  of  California,  for  instance,  arbitrary  east-and-west 
and  north-and-south  lines,  intersecting  at  the  summit  of  Mount 
Diablo  and  called  the  Mount  Diablo  Base  and  Meridian,  have  been 
selected,  and  a  piece  of  land  is  described  as  being  located  so  many 


ACCELERATION.  47 

townships  north  and  east  of  the  Mount  Diablo  Base  and  Meridian, — 
or  N.  and  W.,  or  S.  and  E.,  or  S.  and  W.,  as  the  case  might  be. 

In  Fig.  33,  p.  39,  by  locating  points  A,  B,  C,  etc.,  with  reference, 
to  two  axes  at  right  angles  to  each  other,  we  represented  the  gradual 
changing  of  a  velocity  during  several  seconds,  in  accordance  with  the 
law  expressed  in  formula  5,  p.  42,  v  =  v^-\-o.t.  Now  it  is  true 
generally  that  any  relation  between  two  quantities  can  be  thus 
represented  as  a  line,  either  straight  or  curved,  and  the  character  of 
this  line  will  picture  the  nature  of  the  relation  between  the  two 
quantities. 

With  reference  to  two  such  axes  every  simple  proportion  and 
every  simple  equation  between  two  quantities  can  be  represented  by 
a  straight  line.  The  weight  of  a  given  volume  of  anything  depends 
upon  its  density,  or  W '=  V X  d ;  the  electrical  resistance  of  a  wire 
is  proportional  to  its  length  ;  the  bending  of  a  beam  is  directly 
proportional  to  the  load  upon  it.  Bach  of  these  relations  could  be 
represented  by  a  straight  line. 

More  complex  relations  give  rise  to  curves  of  various  sorts  or  even 
to  broken  lines.  Some  quadratic  equations  produce  ellipses,  others 
circles,  parabolas,  etc.  Temperature  curves  and  barometric  curves, 
showing  the  gradual  rise  and  fall  of  the  thermometer  or  barometer 
throughout  a  day  or  other  interval  of  time,  are  usually  very  irregular. 
A  more  interesting  case  is  the  diagram  on  the  indicator  card  from 
which  the  horse-power  of  an  engine  is  determined,  the  relation  in 
this  case  being  between  the  pressure  of  steam  in  the  cylinder  and  the 
distance  of  the  piston  from  either  end  of  the  stroke.  Fig.  34  shows 
,  ,  such  a  diagram*  taken  from  an  engine 

of  12-inch  stroke  under  a  boiler  pres- 
sure of  75  pounds  per  square  inch,  so 
that  the  total  horizontal  width  of  the 
diagram  corresponds  to  12  inches  of 
stroke  and  the  highest  point  on  the 
Fig.  34-  curves  corresponds  to  about  75  pounds 

pressure  per  square  inch. 

In  every  case,  whether  straight  or  curved,  the  line  pictures  a 
relation  between  two  quantities — velocity  and  time  ;  temperature  and 


Taken  from  a  9.5-inch  x  12-inch  Armington  &  Sims  Standard  Automatic  Cut- 
off Engine  running  200  revolutions  per  minute,  under  boiler  pressure  of  75  pounds 
per  square  inch,  indicating  24.77  H.  P. 


48 


THEORETICAL   MECHANICS. 


time  ;  weight  and  density;  etc.,  —  oue  of  the  quantities  being  a  function 
of  the  other,  or  the  changes  in  one  accompanying  changes  in  the 
other. 


Graphical  Representation  of  Distance  Traveled  by  an 

Accelerated  Body.  To  the  lines  and  points  in  a  diagram 
constructed  upon  co-ordinate  axes  various  meanings  may  be  attached, 
depending  upon  the  idea  to  be  illustrated.  Sometimes,  furthermore, 
the  usefulness  of  the  figure  depends  upon  the  interpretation  of  the 
area  included  within  the  various  lines  and  axes  as  much  as  upon  the 
lines  themselves.  For  example,  in  the  indicator  diagram  from 
which  the  power  and  workings  of  a  steam  engine  are  investigated, 
the  horse-power  is  determined  from  the  area  within  the  curves. 

Reverting  to  Fig.  33,  p.  39,  which  is  here  reprinted 
as  Fig.  35,  it  can  be  shown  that  the  area  Ox-ACy  represents 
the  distance  traveled  by  the  body  referred  to.  Oy  represents 
the  initial  velocity,  and  Oxs  represents  an  interval  of  time  —  3 
minutes.  If  the  velocity  had  continued  equal  to  Oy  (10  miles 
per  hour)  throughout  the  3  minutes,  —  that  is,  if  there  had  been  no 
acceleration,  —  the  distance  traveled  during  that  time  would  have  been 

v  X  /  =  —  ^  —  —  %  mile.  In  the  diagram  v  is  represented  by  Oy 
and  /  is  represented  by  Ox.A.  The  product  of  Oy  and  Ox%  is  the  area 
of  the  parallelogram  Ox^cy,  whence  it  appears  that  the  distance 
traveled  by  any  body  moving  with  a  uniform  velocity  can  be 
represented  by  the  area  of  a  rectangle  of  which  two  adjacent  sides 
represent  the  velocity  and  time,  respectively. 


Fig-  35- 

What  in  the  case  of  accelerated  motion?     If  the  body   under 
consideration,  instead  of  moving  uniformly  at  the  rate  of  10  miles 


ACCELERATION. 


49 


per  hour,  has  an  acceleration  of  2  miles  per  hour,  per  minute, 
increasing  from  Oy  in  the  diagram  to  Oyz  or  x%C  in  the  course  of  3 
minutes  it  is  apparent  that  the  area  Ox^cy  will  represent  only  a  part 
of  the  total  distance.  Study  the  diagram  by  comparison  with  the 

a  f2 
formula  d-  i\t-\ .     It  has  just  been   shown    the  term   v^t   is 

represented  by  the  area  Ox$cyt  which  indicates  what  the  distance 
would  have  been  if  the  velocity  had  remained  constant.  The  term 

—  is  represented  by  the  area  of  the  triangle  ycC*  and  indicates  the 

additional  distance  covered  on  account  of  the  increased  motion.  The 
total  area  Ox^Cy,  therefore,  represents  the  sum  of  the  two,  or  the 
distance  traveled  in  3  minutes  by  a  body  starting  with  a  velocity  of 
10  miles  an  hour  aud  subject  to  an  acceleration  of  2  miles  per  hour, 
per  minute. 

This  method  could  be  used  even  for  variable  accelerations, 
thereby  avoiding  many  complex  computations.  No  matter  how  erratic 
the  changes  from  velocity  Oy  to  velocity  x.AC,  if  they  can  be  put  into 
a  diagram  as  in  Fig.  36,  we  can  easily  determine  the  area  Oxy^y, 


Fig-  36- 

which  represents  the  distance  traveled.  This  area  divided  by  the 
time  Ox,  will  give  the  average  velocity  or  the  average  of  all  the 
possible  vertical  co-ordinates. 

[A  simple  plan  for  determining  the  irregular  area  Oxyly 
would  be  to  cut  out  the  figure  from  a  piece  of  cardboard  and 
compare  the  weight  of  this  irregular  piece  of  cardboard  with  the 
weight  of  a  regular  piece,  or  known  area,  of  the  same  material.] 


*cC  —at  and  yc  =  t  ;  whence  the  area  of  the  triangle  is  equal  to 
'1 
,  as  stated. 


—  '— 


50  THEORETICAL  MECHANICS. 


PROJECTILES. 

The  path  of  a  free  projectile,  undisturbed  except  by 
the  action  of  gravity,  is  always  a  curve.  Whatever  the 
velocity  or  angle  of  projection,  this  curve  always  possesses 
certain  characteristic  features  or  mathematical  relations 
peculiar  to  a  class  of  curves  called  parabolas.  A  projectile 
is  always  subject  to  two  component  motions,  one  of  which 
(the  velocity  of  projection)  is  constant  in  both  magnitude 
and  direction,  while  the  other  (the  gravitational  component) 
is  constant  in  direction  but  of  variable  magnitude. 
If  both  components  were  entirely  uniform  velocities  the 
resultant  path  would  be  a  straight  line  instead  of  a  curve,  ag 
in  Figs.  3  and  4,  pp.  9  and  10.  If  it  were  not  for  the  action 
of  gravity,  a  free  and  unobstructed  body  hurled  in  any  direc- 
tion— horizontally,  obliquely,  or  vertically, — would  move 
on  forever  in  a  straight  line  with  a  uniform  velocity.  The 
effect  of  gravitation,  however,  adds  this  variable  downward 
component,  the  value  of  which  at  any  instant  can  be 
determined  by  the  law  of  falling  bodies.  The  motion  of  a 
projectile  at  any  instant  is,  therefore,  the  resultant  of  two 
components, — one  a  uniform  velocity  in  the  direction  of 
projection,  and  the  other  a  uniformly  accelerated  velocity 
vertically  downward. 

For  instance,  suppose  a  ball  is  rolled  along  a  platform 
OA,  Fig.  37,  with  a  uniform  velocity,  say  40  feet  per  second. 
If  the  platform  remains  at  rest,  the  position  of  the  ball  at  the 
end  of  successive  seconds  will  be  at  points  i,  2,  3,  etc.  But  if 
the  platform  is  allowed  to  fall  freely,  at  the  end  of  i  second 
it  will  be  in  position  O^A^  16.1  feet  below  OA,  and  the 
ball  will  be  at  i'.  At  the  end  of  2  seconds  the  ball  will  be 
to  2',  64.4  feet  below  2,  etc. 


ACCELERATION.  51 

It  should  be  noted  that  this  diagram  (Fig.  37)  is  very 
different  from  some  of  the  graphical  representations 
previously  used  ;  in  this  case  the  horizontal  and  vertical 
magnitudes  represent  distances  and  not  velocities,  and  the 
curved  line  is  not  the  resultant  velocity,  but  the  actual  path 
traversed.  The  resultant  velocity  at  any  instant  is  a  tangent 
to  the  curve  at  the  point  where  the  body  happens  to  be  at 
that  instant.  To  compute  this  resultant  we  must  combine 


Fig-  37- 

the  fixed  horizontal  component  and  the  computed  vertical 
component  for  the  given  instant.  These  velocities  may  be 
represented  graphically  on  any  scale  desired,  independently 
of  the  scale  of  distances  used  in  constructing  the  path.  At 
point  O,  for  example,  the  horizontal  component  is  40  feet 
per  second  and  the  vertical  component  is  zero  ;  the  resultant 
is  40  feet  per  second  horizontally.  At  the  very  next 
instant — the  merest  fraction  of  a  second — it  is  something 
different.  At  point  i'  the  horizontal  component  is  40  feet 
per  second  and  the  vertical  component  is  32.2  feet  per 


52  THEORETICAL   MECHANICS. 

second;  the  resultant  is  \/4O2  -f-  32. 22,  at  an  angle  8  = 
tan"1  32.2-^-40.*  At  point  3'  the  horizontal  component  is 
still  40  feet  per  second  and  the  vertical  component  is  96.6 
feet  per  second ;  the  resultant  is  \/4o2  -f  96. 62,  at  an 
angle  8  =  tan"1  96. 6 -=-40. 

The  resultant  of  two  uniform  components  is  constant  in  both 
magnitude  and  direction  ;  uniform  motion  in  a  circle  is  constant  in 
rate  but  continually  changes  direction  ;  the  motion  of  a  projectile 
changes  in  both  magnitude  and  direction. 

Another  interesting  contrast  is  had  by  comparing  the  motion  of 
a  projectile  with  the  resultant  motion  of  a  point  on  the  perimeter  of 
a  carriage  wheel  describing  a  cycloidal  path,  as  in  Fig.  29.  A  point 
on  the  carriage  wheel  is  subject  to  two  component  motions,  both  of 
which  remain  constant  in  magnitude,  while  in  the  matter  of  direction 
one  component  remains  always  parallel  to  the  ground  and  the  other 
(the  tangential  component)  continually  changes  direction  as  the 
wheel  revolves.  The  point  on  the  wheel,  therefore,  has  both 
components  constant  in  magnitude  and  one  changing  in  direction  ; 
a  projectile  has  both  components  constant  in  direction,  and  one 
changing  in  magnitude. 

In  the  baseball  problem,  p.  30,  we  assumed  that  the  ball 
maintained  a  horizontal  direction  like  the  carriage  wheel.  If  we 
allow  for  the  action  of  gravity,  then  it  will  assume  a  parabolic  path 
like  any  projectile,  and  the  motion  of  any  point  in  it  will  be  the 
resultant  of  three  components:  (i)  the  velocity  of  projection, 
constant  in  all  respects ;  (2)  the  tangential  component  due  to  the 
rotation  of  the  ball,  constant  only  in  magnitude ;  (3)  the  gravita- 
tional component,  constant  only  in  direction. 

Examples : 

1.  (a)  Construct  a  diagram  of  the  path  of  a  projectile  hurled 
horizontally  with  a  velocity  of  100  feet  per  second  from  a  height  of 
579.6  feet. 

INSTRUCTIONS  :  For  graphical  representation  use  a  scale  of 
i  inch  =  loo  feet.  First  prepare  a  table  of  distances,  such  as  the 


*tan-i  32.2-^40  means  "the  angle  whose  tangent  is  32.2^-40." 


ACCELERATION. 


53 


following,  in  which  the  time  refers  to  the  number  of  seconds  after 
starting,  and  the  corresponding  horizontal  and  vertical  distances 
traveled  are  reduced  to  scale. 


Time.              Horizontal  Distance. 

Vertical  Distance. 

[Reduced  to  Scale.] 

[Reduced  to  Scale.] 

/  =  0.5  seconds        50  feet      0.5  inches 

4.0  feet          o  040  inches 

i.b 

IOO 

I.O 

16.1 

0.161 

i-5 

150 

i  5 

36.2 

0.362 

2.0 

2OO 

2.O 

64.4 

0.644 

2-5 

250 

2-5 

100.6 

i.  006 

3-o 

300 

3-o 

144-9 

1.449 

4.0 

400 

4.0 

257.6 

2.576 

5-o 

500 

5-0 

402.5 

4.025 

6.0 

600 

6.0 

579-6 

5-796  , 

Prepare  a  cross-section  sheet  divided  into  one-inch  squares  with 
extra  vertical  lines  at  half-inch  intervals  where  required  by  table. 
Beginning  at  upper  left-hand  corner  as  the  origin,  locate  points 
determined  by  values  reduced  to  scale  in  table,  and  connect  successive 
points,  starting  from  origin.  In  preparing  diagram  use  a  sharp,  hard 
pencil. 

(b)  Compute    the   resultant   velocity  of  this  body    at  the 
instant   of   striking  the    ground,  giving  the    direction  of    the 
resultant  as  well  as  its  magnitude. 

(c]  What  was  its  resultant  velocity  at  the  end  of  the  third 
second^ 

2.  A  body  is  projected  with  a  velocity  of  20  feet  per  second  at  an 
angle  of  jj0  above  the  horizon.  Construct  a  diagram  of  its  path. 


INSTRUCTIONS  :  Use  a  scale  of  i  inch  =  5  feet.  Tabulate  values 
of  distances  traveled  in  direction  of  projection,  and  the  falling 
distances,  for  successive  periods  of  time  changing  by  i/io  second, 
after  manner  of  last  example.  Trace  the  path  up  to  the  end  of  1.5 
seconds.  In  ruling  the  cross-section  sheet  draw  the  usual  vertical  base 
line,  but  construct  the  second  reference  line  at  the  proper  angle  of 
35°  above  the  horizon.  On  the  latter,  measure  off  distances  of  0.4 
inch  (representing  distance  of  2  feet  traveled  in  that  direction  during 
each  o.i  second),  and  through  each  of  these  points  draw  a  vertical 
line.  In  locating  points  determined  by  the  values  in  table,  count  the 
uniform  values  along  the  inclined  reference  line,  and  from  each  of  the 


54 


THEORETICAL  MECHANICS. 


successive  points  marked  on  that  line,  measure  the  falling  distance 
for  the  time  interval  referred  to. 

The  path  should  resemble  the  curve  in  Fig.  38. 

3.     Prove: 

(/)  A  body  hurled  horizontally  from  any  altitude  reaches  the 
.  ground  in  the  same  time  as  if  it  had  been  merely  dropped  and 
allowed  to  fall  vertically. 

(ii)  A  body  projected  at  any  angle  reaches  the  ground  in  the 
same  time  as  if  it  had  been  projected  vertically  with  a  velocity 
equal  to  the  vertical  component  of  the  actual  velocity  of  projection. 

Elevation  and  Range  of  a  Projectile.  The  angle  of 
projection  above  the  horizon  is  called  the  Elevation.  The 
horizontal  distance  OA,  Fig.  38,  is  called  the  Range.  The 


Fig,  38- 

highest  point  of  the  path  is  called  its  Vertex,  of  which  BC 
is  the  height. 

The  Range  and  Height  of  Vertex  are  easily  determined 
by  computation.  For  this  purpose  resolve  the  velocity  of 
projection  into  its  horizontal  and  vertical  components.  A 


ACCELERATION.  55 

body  projected  with  a  velocity  Vp,  at  an  angle  /?,  will  rise  to 
the  same  vertical  height  and  in  the  same 
time  as  if  it  were  thrown  vertically 
upward  with  a  velocity  equal  to  com- 
ponent a.  Now,  we  have  already  learned 
how  to  find  the  height  to  which  a  body 
will  rise  if  hurled  upward  with  a  given 

velocity,  and  the  time  that  will  elapse  before  it  returns  to 

the    ground.       This    time    multiplied    by    the    horizontal 

component  will  give  the  range. 

For   example,    the    Range    and    Height    of  vertex    in 

example  2,  p.  53,  would  be  determined  as  follows  : 

Vp    =    20 

a  =  vp  sin  35°=   20  X  .5736  =    n-472 
b  -  zycos'-35°=  20  X  .8192  =  16.384 

An  upward  vertical  velocity  suffers  an  acceleration  of 
—  32.2  feet  per  second,  per  second  on  account  of  gravity, 
whence  this  body  will  reach  its  highest  point  in 

t-       '-  —  •-  =  o.  3  5  6  second  , 
at  an  altitude  of 


h-       '~~     X.  356  =  2.042  feet. 


To  find  the  range,  R,  multiply  the  horizontal  component 
of  Vp  by  2  t,  the  time  that  elapses  while  the  body  moves  to 
the  highest  point  and  back  to  the  horizontal. 
R  -  2  X  .356  'X  16.384  =  11.665 


Examples  : 

1.  From  your  diagram  constructed  for  example  2,  p.  jj>,  determine 
the  Range  and  Height  of  Vertex  by  scale,  and  compare  with  the  above 
results. 

2.  A  body  is  projected  with  a  velocity  of  20  feet  per  second  at  an 
elevation  of  45°.      Construct  a  diagram  of  its  path  up  to  the  end  of 


56  THEORETICAL   MECHANICS. 

7.5  seconds.  Determine  its  height  of  vertex  and  range  by  scale  and  by 
computation. 

•3.  Do  likewise  for  a  projectile  hurled  with  a  velocity  of  21 
miles  per  hour  at  an  elevation  of  jj°. 

4.  A  shell  is  fired  from  one  of  the  pneumatic  guns  on  Presidio 
Cliff  with  a  velocity  of  30.000  feet  per  minute  at  an  elevation  of  4.7°. 
If  the  gun  is  400  feet  above  sea  level,  compute  the  time  at  which  the 
shell  will  strike  the  water,  the  horizontal  distance  from  the  bottom  of 
the  cliff  to  the  striking  point,  and  the  greatest  altitude  reached. 

Maximum  Range  of  a  Projectile.  If  it  were  not  for  the 
resistance  of  the  air  a  projectile  hurled  with  any  velocity 
would  have  a  maximum  range  when  the  angle  of  elevation 
.is  45°.  In  Fig.  38, 

OA  =  OD  cos  ft, 

where  ft  is  the  angle  of  elevation.      But 

OD  =  vp  4, 

in  which  ta  is  the  time  to  reach  A,  or  the  time  that  elapses 
while  the  body  is  moving  to  the  highest  point  and  back  to 
the  horizontal.  And 

_     z>sin£ 

i' a  ~ —    *•  —  ' 

32.2 

whence 

^  A  o     2  VP~  sin  /3cos  ft 

OA  =  vp  ta  cos  ft  =  —p—  —  • 

32.2 

But 

2  sin  ft  cos  ft  =  sin  2  ft, 
whence 

OA  =  ^-  sin  2/3. 
32.2 

Since  z^,2-;- 32. 2  is  a  constant,  the  range  OA  is  a  function 
of  sin  2  ft,  and  hence  OA  has  its  maximum  value  when 
sin  2  ft  is  greatest.  The  greatest  value  that  the  sine  of  any 
angle  can  have  is  the  sine  of  90°,  or  one.  Putting  sin  2  ft  =  i , 
2  ft  will  then  be  90°,  or  ft  =  45°.  Therefore,  the  range  OA  is 
greatest  when  ft=  45°,  whatever  the  velocity  of  projection. 


SECTION  IL 
STATICS. 


CHAPTER  I. 
FORCE.        MASS. 

In  Section  I,  we  learned  that  the  subject  of  Kinematics  is 
based  upon  considerations  of  space  and  time.  In  all  the 
remaining  branches  of  mechanics,  including  the  present 
subject  of  Statics,  we  must  deal  with  a  new  idea  ;  viz., 
Force, — a  clear  and  correct  conception  of  which  is  not 
easily  grasped. 

In  common  usage  the  word  "force"  has  been  widely 
applied  and  greatly  abused.  In  the  preceding  chapters  we 
have  avoided  using  it,  substituting  for  it  such  words  as 
"influence,"  and  "cause,"  in  order  that  we  might  begin  to 
use  it  at  a  time  when  we  can  consider  its  proper  scientific 
meaning. 

If  two  bar  magnets,  suspended  so  as  to  swing  freely,  are 
placed  near  each  other,  they  will  exhibit  attraction  or 
repulsion,  depending  upon  the  character  of  adjacent  poles. 
Without  stopping  to  theorize  concerning  the  remote  cause 
of  this  influence  between  the  two  magnets,  we  can  accept 
the  facts  that  there  is  an 'action  in  which  both  magnets 
participate — a  mutual  action,  and  that  it  results  in  motion 
or  exhibits  itself  through  the  motion  it  produces. 


58  THEORETICAL  MECHANICS. 

This  mutual  action  between  the  two  given  masses  is 
called  a  Force. 

An  electrified  rod  of  glass,  ebonite  or  other  non-conductive 
material,  or  even  an  insulated  conductor  when  similarly 
electrified  by  rubbing  or  otherwise,  will  attract  a  stick  or 
rod  of  metal  suspended  so  as  to  swing  freely.  In  fact,  all  the 
familiar  phenomena  of  electrical  attraction  and  repulsion  are 
instances  of  an  action  between  two  masses  quite  analogous 
to  the  magnetic  action  above  referred  to.  The  remote 
cause  of  electrical  attraction  and  repulsion  may  be  very 
different  from  the  cause  of  the  analogous  magnetic 
phenomena,  but,  nevertheless,  in  each  case  the  ultimate 
effect  is  an  action  between  the  two  bodies  concerned  in  the 
transaction,  which  moves,  or  tends  to  move,  both  of  them. 
We  may  have,  therefore,  a  magnetic  force  or  an  electrical 
force. 

If  an  elastic  cord  or  a  spiral  spring  be  elongated,  and  a 
ball  or  other  mass  be  fastened  at  each  end,  the  cord  or  spring 
will  contract  again  as  soon  as  freed,  and  will  even  move  the 
two  inert  masses.  The  rebound  of  a  rubber  ball  is  due  to 
the  same  cause.  In  such  cases  the  moving  influence  is  an 
action  of  some  sort  between  the  particles  of  the  rubber  or  ot 
the  material  in  the  spring.  But  it  is  an  action  which 
produces  motion,  and  hence  is  a  force.  It  differs,  however, 
from  an  action  between  two  distinct  bodies  separated  by  an 
appreciable  distance,  as  in  the  case  of  the  bar  magnets, 
because  if  the  rubber  band  were  stretched  far  enough  to  break 
it  there  would  be  no  perceptible  attraction  between  the  two 
parts.  It  is  some  manifestation  of  cohesion  between  the 
individual  particles  or  molecules  of  the  rubber,  and  the  sum 
total  of  these  molecular  actions  is  what  we  observe  as  the 
total  force  exerted  by  the  rubber.  It  may  be  designated  as 
a  molecular  force,  or  an  elastic  force,  or  by  means  of  any 


FORCE.  59 

other   adjective   that  will  identify  it  through  its  origin  or 
through  any  of  its  characteristics. 

But  the  commonest  exhibition  of  force  is  the  action  of 
gravity,  with  which  we  are  all  familiar  in  a  general  way.  To 
our  senses  this  action  is  evidenced  through  the  phenomena 
of  attraction  between  large  masses,  such  as  the  general  mass 
of  the  earth  and  objects  on  its  surface,  though  the  accepted 
theory  of  gravitation,  which  will  be  considered  in  a  later 
paragraph,  gives  it  a  molecular  origin.  The  theories  of 
electrical  and  magnetic  forces  would  put  these  also  on  a 
similar  molecular  basis.  In  all  cases,  however,  whether  the 
size  of  the  body  or  the  quantity  of  matter  under  considera- 
tion be  great  or  small,  the  force  exhibits  itself  as  an  action 
between  two  masses. 


Definition :— A  Force  is  a  mutual  action  of  attraction  or 
repulsion  between  two  masses. 

The  masses  themselves  may  move  under  this  influence,  or  the 
force  may  be  employed  to  move  some  other  mass,  as  the  stirrups  in 
which  the  magnets  are  suspended,  or  the  two  balls  attached  to  the 
rubber  cord,  or  an  engine  driven  by  the  expansive  force  of  steam. 


Resistance.  If  a  person  lifts  a  weight  from  the  ground 
the  force  exerted  is  a  muscular  contraction  (similar  to  the 
contraction  of  a  rubber  band),  which  tends  to  pull  the 
shoulder  downward  and  the  hand  upward.  If  the  weight 
is  raised  we  say  that  the  resistance  of  gravity  has  been 
overcome.  For  the  time  being  we  cease,  or  forget,  to  think 
of  gravity  as  an  active  agent ;  we  regard  the  muscular 
force  as  the  action  and  the  force  of  gravity  as  a  resistance, 
merely.  But  reverse  the  thought :  if  it  were  not  for  the 
resistance  of  the  hand  holding  it  the  weight  would  move  to 
the  earth,  and  so  we  think  of  the  hand  as  resisting  the 


60  THEORETICAL  MECHANICS. 

action  of  gravity.  When  action  is  pitted  against  action, 
force  against  force,  in  this  manner,  it  may  suit  our 
convenience  to  regard  either  as  a  resistance  to  the  other. 

When  a  force  acts  upon  a  body  to  bend  it,  or  stretch  it,  or 
compress  it,  or  otherwise  deform  it,  there  is  occasioned  in 
the  body  a  resistance  which  had  no  existence  until  called 
into  play  by  the  action  of  the  force  itself.  This  is  a  case 
somewhat  different  from  two  entirely  independent  forces 
counteracting  each  other.  For  instance,  a  body  resting  on 
a  table  is  prevented  from  falling  to  the  ground  because  the 
table  resists,  or  stands  against,  the  action  of  gravity.  The 
attraction  between  the  weight  and  the  earth  compresses  the 
table  and  crowds  its  molecules  together.  The  molecules 
resist  this  displacement  from  their  normal  positions  and 
exert  among  themselves  an  expansive  force  or  repulsion, 
which  causes  the  table  to  assume  its  original  form  and 
dimensions  when  the  load  is  removed.  Thus  it  appears  that 
the  elasticity  of  a  body  may  be  called  into  play  to  furnish  a 
force  that  will  resist  an  external  force.  Under  such 
conditions  the  body  is  said  to  be  subject  to  a  Stress.  A 
stress  may  be  a  Pressure,  or  a  Tension^  or  a  Shearing  Stress. 

A  body  on  a  rough  surface  offers  a  frictional  resistance 
to  any  force  tending  to  move  it  along  the  surface. 
This,  likewise,  is  always  a  passive  resistance  rather  than  a 
counter-action  ;  it  can  have  no  existence  until  called  into 
play  by  the  action  of  gravity  or  some  other  force,  and  it  is 
merely  a  recipient  of  such  action. 

These  are  typical  illustrations  of  the  phenomena  and 
considerations  to  be  dealt  with  in  Statics.  A  force  being  a 
mutual  action  of  attraction  or  repulsion  between  two  masses, 
it  always  tends  to  produce  motion.  If  it  is  counteracted,  or 
counterbalanced,  in  any  manner  whatsoever,  it  gives  rise  to 
statical  conditions. 


FORCE.  6 1 

Action  and  Reaction.  The  following  law,  from  the  time 
that  it  was  first  enunciated  by  Newton;  viz.,  "To  every 
action  there  is  an  equal  and  opposite  reaction,"  has  proved 
more  or  less  misleading  to  beginners.  Let  us  investigate 
some  of  the  causes  leading  to  a  misunderstanding  of  this 
simple  statement.  Where  there  is  but  one  force  under 
consideration — a  mutual  or  dual  action  between  two 
masses,  the  application  of  the  idea  of  "action  and  reaction" 
is  quite  simple.  If  two  bodies,  A  and  B,  are  connected  by  an 
elastic  cord,  the  contraction  of  the  cord  pulls  equally  on 
A  and  B,  in  opposite  directions.  Or,  if  the  force  is  not 
exerted  by  an  elastic  cord,  but  A  and  B  attract  each  other  by 
gravity  or  otherwise,  then  this  dual  or  mutual  action  may 
be  regarded  as  a  simultaneous  operation  of  two  separate 
reciprocal  actions — A  upon  B,  and  B  upon  A,  and  whichever 
we  choose  to  mention  first,  we  may  call  the  "action,"  and 
the  other  is  its  reciprocal  action,  or  its  "reaction."  To 
this  extent  the  idea  of  ' '  action  and  reaction  ' '  simply  asserts 
the  dual  nature  of  a  force. 

It  is  in  cases  where  one  force  is  balanced  by  another, 
such  as  lifting  a  weight  from  the  ground,  that  the  complex- 
ities and  difficulties  arise.  We  have  already  stated  that  in 
lifting  the  weight  the  muscular  contraction  of  the  arm  pulls 
down  on  the  shoulder  and  upward  on  the  hand — equally 
and  in  opposite  directions,  like  the  elastic  cord, — giving  rise 
to  two  reciprocal  operations,  either  of  which  may  be  taken 
as  the  action  and  the  other  as  the  reaction.  This  is  still 
a  consideration  of  a  single  force,  and  the  idea  is  not  yet 
confused.  But,  suppose  we  say  that  the  downward  tendency 
of  the  weight  is  balanced  by  the  upward  pull  of  the  hand, 
as  already  stated  in  speaking  of  resistances:  can  we  regard 
these  two  opposing  influences  as  a  case  of  ' '  action  and 
reaction  ?"  It  is  here  that  the  misunderstanding  originates, 


62 


THEORETICAL   MECHANICS. 


—when  the  idea  of  "action  and  reaction"  is  extended  to 
cover  this  class  of  cases  which  we  have  just  termed  resistances. 
It  is  true  that  these  two  opposing  influences  are  equal 
and  opposite,  because  they  balance  each  other,  but  they  can 
not  produce  motion  and  do  not  constitute  a  force  according 
to  our  definition  ;  on  the  contrary  each  is  a  part  or  member 
of  two  different  forces  counteracting  each  other.  That  is,  in  the 


I.    FORCE  OF  GRAVITY. 

Weight  grasped  by 
hand  is  pulled  downward 
towards  earth. 


These  arrows  in- 
dicate the  Action 
and  Reaction  in 
volrcd  in  the  Force- 
of  Gravity. 


Earth  is  attracted  up- 
ward towards  weight 
and  against  foot,  the  lat- 
ter pressure  being  trans- 
mitted to  the  shoulder. 


These   diagonal    lines 
how  how  the  two  forces 


balance  each  other,  there 
being  two  pairs  of  equil- 
ibrations. 


\/ 


II.  MUSCULAR  FORCE. 

Shoulder  is  pulled 
downward  by  muscle 
attached  to  hand,  and 
this  pressure  is  rans- 
mitted  through  the  fout 
against  the  earth. 


These  arrows  in- 
dicate the  Action 
and  Reaction  in- 
volved in  the  Mus- 
cular Force 


Hand  grasping  weight 
is  pulled  upward  by 
muscle  attached  to 
shoulder. 


Fig-  39- 


case  under  consideration,  we  have  one  of  the  members  of  a 
second  force — the  upward  muscular  action  on  the  hand — 
balancing  the  downward  action  of  gravity  on  the  weight. 
But  this  is  only  half  the  transaction  ;  what  becomes  of  the 
remaining  member  of  each  of  these  two  forces?  They 
must  also  balance  each  other,  if  the  equilibrium  between  the 
two  forces  is  to  be  complete.  That  is,  if  the  upward 


FORCE.  63 

muscular  pull  on  the  hand  balances  the  downward  pull  of 
gravity  on  the  weight,  then  the  downward  muscular  pull  on 
the  shoulder  must  balance  the  upward  pull  of  gravity  on  the 
earth.  "The  actions  and  reactions  of  the  two  forces 
equilibrate  in  pairs."  Owing  to  the  dual  nature  of  each 
force  it  requires  a  double  transaction  of  resistances  in  order 
that  the  two  forces  may  balance  each  other.  In  practice  we 
do  not  generally  take  cognizance  of  more  than  one  of  these 
equilibrations  at  a  time,  but  the  other  occurs,  nevertheless. 

These  things  are  illustrated  by  diagram  in  Fig.  39, 
which  shows  : 

First — That  each  of  the  given  forces — gravity  and 
muscular  contraction — involves  an  action  and  its  reaction. 

Second — That  these  two  forces  balance  each  other 
completely,  giving  two  pairs  of  resistances  or  equilibrations. 

Similarly,  a  weight  resting  on  a  table  is  sometimes 
cited  as  an  instance  of  action  and  reaction.  The  weight 
pushes  downward  on  the  table  and  the  table,  it  is  asserted, 
' 'reacts"  upon  the  weight.  As  a  matter  of  fact  the  prime 
active  agency  in  the  case  is  the  force  of  gravity  between 
the  weight  and  the  earth,  so  that  if  we  start  with  the 
downward  tendency  of  the  weight  and  call  it  the  ' 'action" 
then  its  real  reaction  is  the  upward  tendency  of  the  earth. 
As  each  of  these  tendencies  is  overcome  by  the  interposition 
of  the  table,  we  have  two  pairs  of  equilibrations — the  top  of 
the  table  versus  the  weight,  and  the  bottom  of  the  legs 
versus  the  earth.  But  to  say  that  the  table  reacts  against 
the  weight,  and  the  legs  against  the  earth,  is  an  objectionable 
use  of  the  idea  of  action  and  reaction.  It  is  confounding  a 
counter-action  with  a  r^-action.  The  action  and  reaction,  or 
force,  between  the  weight  and  the  earth  is  not  destroyed  or 
even  suspended  by  the  intervention  of  the  table,  but  merely 
restrained  from  producing  motion. 


64  THEORETICAL   MECHANICS. 

Composition  and  Resolution  of  Forces.  A  force  can  be 
represented,  in  both  magnitude  and  direction,  by  a  straight 
line.  Component  forces  may  be  combined  geometrically  to 
determine  their  resultant,  precisely  after  the  manner  of  the 
"  composition  of  velocities."  Some  simple  applications  of 
the  graphical  representation  of  forces  will  be  employed 
incidentally,  without  further  explanation,  thoughout  the 
subject  of  Statics. ' 

Law  of  Gravitation.  As  already  stated,  the  action  of 
gravity  is  the  most  familiar  example  of  force.  Electric  and 
magnetic  forces  are  seldom  dealt  with  in  the  study  of 
theoretical  mechanics,  the  problems  and  illustrations  usually 
referring  to  weights,  and  to  pressures,  tensions,  etc.,  due 
to  weights.  The  familiar  facts  of  gravity  are  these  : 

That  the  weight  of  a  body  depends  upon  the  amount  of 
material  in  it ;  the  weight  is  directly  proportional  to  the 
mass. 

That  forces  of  all  kinds, — electric,  magnetic,  muscular, 
etc., — are  usually  balanced  by  weights,  and  hence  are 
measured  in  gravitational  units,  such  as  pounds,  ounces, 
grams,  kilograms,  etc. 

That  weight  acts  vertically  downward  towards  a  point 
at  or  near  the  center  of  the  earth. 

That  the  weight  of  a  given  body  depends  somewhat 
upon  its  distance  from  the  center  of  the  earth,  being  greater 
at  the  poles  than  at  the  equator,  and  greater  near  the  sea 
level  than  at  high  altitudes. 

That  the  rotation  of  the  earth  diminishes  the  weight  of 
bodies,  with  greatest  effect  at  the  equator  and  reducing  to 
zero  at  the  poles. 

While  these  general  considerations  may  be  clear  in  a 
general  way,  yet  a  proper  appreciation  of  the  Law  of 
Gravitation  requires  considerable  practice  in  the  use  of  the 
so-called  mathematical  law  of  inverse  squares,  as  applied  to 
problems  of  celestial  mechanics,  such  as  planetary  attractions. 


FORCE.  65 

Mathematical   Expression   of  the   Law  of  Gravitation. 

The  lyaw  of  Gravitation  asserts  that  between  every  pair 
of  particles  in  the  universe  there  is  a  mutual  attraction, 
which  varies  (i)  directly  as  the  product  of  the  masses  of  the 
particles,  and  (2)  inversely  as  the  square  of  the  distance 
between  them. 

(  i  )  If  A  represents  the  force  of  attraction  between  two 
particles  of  matter,  m  and  m\  then  (disregarding  distance 
for  the  present,  and  using  the  sign  of  variation  to  avoid  the 
necessity  of  selecting  units  of  force  and  mass),  according  to 
the  first  part  of  the  law, 

A  oc  m  X  m'  .  * 

The  particles  m  and  m'  may  be  alike,  or  they  may  be  as 
different  as  we  please  ;  they  may  even  be  magnetized  or 
electrified,  and  thus  have  a  second  attraction  (or  repulsion) 
entirely  independent  of  gravity.  Their  gravitational 
attraction  may  be  represented  by  diagram,  thus  : 

»>o  —  »  *  <  —  ox 

Fig.  40. 

If  m  be  replaced  by  two  particles  of  the  same  kind,  or 
by  a  single  particle  of  twice  the  mass,  the  attraction  will  be 
doubled,  thus  : 

i*  1=0 


Fig.  41. 

If  both  m  and  m'  be  doubled  the  attraction  becomes  four 
times  as  great,  (the  distance,  of  course,  being  assumed  to 
remain  unchanged),  thus  : 


Fig.  42. 

If  m  be  increased   to  $m  and  m'  to  $m'  ,  the  attraction 
will  become  15  A  ;  etc. 


The  symbol  oc  means  "varies  with. 


66  THEORETICAL   MECHANICS. 

(2)  If  the  two  original  particles  be  not  changed  in 
mass,  but  are  moved  to  different  distances  from  each  other, 
then  as  this  distance  d  is  changed 

Aa:±. 
If  the  attraction  is  A  at  distance  d  (Fig.  43), 


at  distance  2d  (Fig.  44)  it  will  be  -— . 

4 

O  -*-  O 

Fig.  44. 

A 

At  distance  $d  it  will  be  -— ;  etc.       If  d  be  diminished  to 
-\rd,  it  will  be  $A. 

If  the  masses  and  distance  both  change  simultaneously, 
what  is  the  result  ?  For  instance,  if  m  be  doubled  and  m' 
tripled,  and  the  distance  doubled  at  the  same  time,  the 
increased  masses  will  multiply  the  attraction  6  times,  but 
the  increased  distance  will  diminish  the  result  to  one-fourth 
of  what  it  would  have  been  if  the  distance  had  remained 
unchanged.  The  result  will  be  an  attraction  f-  as  great 
as  the  original  A  between  m  and  m'  at  distance  d.  That  is, 

mm' 

A   OC        j^ 


The  attraction  between  two  large  spherical  masses  of 
uniform  density  is  the  same  in  effect  as  if  the  entire  material 
in  each  mass  were  condensed  into  its  central  particle.  For 


FORCE.  67 

instance,  in  computing  the  attraction  between  objects 
and  the  earth,  we  consider  the  distance  to  the  center  of  the 
earth  and  not  to  the  surface. 

Examples: 

1.  A  body,  or  aggregate  of  particles,  has  a  mass  of  8  units,  and 
a  second  body  has  a  mass  of  /j  units.  At  a  distance  of  12  units  from 
each  other,  these  bodies  exhibit  a  gravitational  attraction  equal  to 
30  grams.  Express  the  attraction  between  each  of  the  following 
Pairs  of  bodies  at  the  distances  specified : 

Mass  of  each  of  given  Distance  between 

bodies.  bodies. 

I  II 

5  units  6  units  24  units 


5 
2 

(iv)  30 

(v)  40 

(vi)  25 


6 

10 

3 

7 

13 


48 
6 

2 

8 
18 


2.  The  attraction  between  two  masses,  m  and  2m,  at  distance  d 
from  each  other,  is  equal  to  A.     Another  pair,  of  magnitudes  2m  and 

3m,  are  placed  at  a  distance  3d  from  each  other  ;  what  is  the  attraction 
between  them,  expressed  in  terms  of  A  f 

3.  A  body  on  the  earth1  s  surface  weighs  one  pound. 

(a)  What  will  it  weigh  4000  miles  above  the  earth1  s  surface? 
(&)  What  will  it  weigh  8000  miles  above  the  earth's  surface  f 
(c]  What  will  it  weigh  6000  mi/es  above  the  earth's  surface  ? 

It  is  assumed   that  the   body   is   always   weighed   on  a  spring 
balance  graduated  at  the  earth's  surface.     Why? 

4.  The  earth  is  said  to  be  slowly  cooling  and  contracting.    How 
does  this  affect  the  weights  of  bodies  on  the  earth's  surface  f       Why  ? 
What  would  be  the  effect  if  the  earth  were  expanding?     Why  f 

5.  Assuming  that  the  mass  of  the  Moon  is  ¥V  that  of  the  earth, 
is  there  a  point  between  them  at  which  a  body  would  have  no  weight? 
If  so,  locate  it. 

6.  A  spherical  body  weighs  10  pounds  at  a  given  place  on  the 
earth's  surface.       Find  the  weight  at  the  same  place  of  a  second 
sphere  of  twice  the  radius  and  three  times  as  dense  as  the  first  body. 


68  THEORETICAL   MECHANICS. 

7.  A  given  body  weighs  one  pound  on  the  earth's  surface. 

(a)  What  will  it  weigh   on  another  planet  of  the    same 
density  and  whose  mass  is  27  times  that  of  the  earth  ? 

(b]  What  will  it  weigh  on  still  another  planet  of  the  same 
density,  but  the  mass  of  which  is  -fa  that  of  the  earth  ? 

8.  A  spherical  body  weighs  100  pounds  on  the  earth' s  surface. 
Find  the  weight  of  a  second  body  of  twice  the  radius  and  twice  as 
dense,  situated  on  the  surface  of  a  planet  of  three  times  the  earth's 
radius,  but  only  half  as  dense  as  the  earth. 

Weight  of  a  Body  Beneath  the  Earth's  Surface.  It  has 
been  shown  that  a  body  carried  above  the  earth's  surface 
loses  weight  inversely  as  the  square  of  the  distance  from  the 
earth's  center,  but  we  should  not  infer  from  this  that  a  body 
would  gain  weight  if  it  could  be  carried  in  the  opposite 
direction,  towards  the  earth's  center.  On  the  contrary,  it 
is  obvious,  that  if  the  earth  were  a  sphere  of  uniform  density, 
a  particle  at  the  center  would  be  attracted  equally  in  all 
directions,  and  furthermore,  it  can  easily  be  demonstrated 
that  a  body  in  transit  from  the  surface  to  the  center  would 
lose  its  weight  uniformly  with  the  distance.  The  weight  of 
such  a  body  would  vary  directly  as  the  distance  from  the 
center,  and  not  inversely  as  the  square  of  the  distance. 
While  this  demonstration  involves  practically  nothing  more 
than  the  law  of  gravitation,  it  gives  rise  to  the  application 
of  the  law  of  inverse  squares  to  the  following  important 
proposition  : 

A  body  placed  at  any  point  within  a  hollow  spherical 
shell  of  uniform  density  is  at  perfect  equilibrium  as  regards 
the  attraction  of  the  shell. 

In  Fig.  45,  let  P  be  a  particle  at  any  point  within  the 
shell  shown  in  section  in  the  diagram.  Let  PB  and  PC 
represent  elements  of  a  right  circular  cone.  If  the  surface 
of  this  cone  were  continued,  it  would  cut  out  of  the 


FORCE.  69 

spherical  shell  a  concave  disc  represented  in  section  as 
AB.  BP  and  CP  continued  will  form  elements  of  a  second 
and  similar  right  circular  cone,  which  would  cut  from  the 
shell  a  second  disc  of  section  AlBl.  (We  shall  speak  of 
these  sections  in  place  of  the  area  which  they  represent). 


By  Prop.  VI,  Book  IX  of  Euclid,  (Wells'  Geometry, 
P-  35 5) i  the  areas  of  the  bases  BC and  BlCl  of  these  cones 
are  to  each  other  as  PD  2  is  to  PD*.  As  the  vertex  angle 
of  the  cone  diminishes,  the  bases  BC  and  BlCl  approach  the 
areas  AB  and  A1BIJ  respectively,  as  limits.  Therefore,  for 
any  small  area,  as  the  limiting  area  AB,  there  is  a 
corresponding  area  A1B1  so  situated  that 


area  AB         PH* 


area  A  1jB1      />// 


Under  these  circumstances  it  is  obvious  that  the 
attraction  between  the  particle  at  P  and  the  mass  of  AB  is 
iust  equal  and  opposite  to  the  attraction  between  the 


70  THEORETICAL   MECHANICS. 

particle  at  P  and  the  mass  of  A^B^.  For  instance,  if  the 
mass  included  within  the  area  Al£l  is  four  times  as  great 
as  the  mass  of  AB,  it  can  be  so  only  because  the  square  of 
the  distance  PH^  is  also  four  times  the  square  of  PH ;  the 
predominance  of  mass  Al^l  is  exactly  neutralized  by  its 
greater  distance  (squared). 

If  it  were  not  that  this  relation  of  dimensions  of  the 
spherical  shell  happens  to  coincide  in  effect  with  the  law  of 
inverse  squares  as  applied  to  attractions,  a  particle  would 
not  be  at  equilibrium  at  any  point  within  the  shell,  except 
at  the  very  center  ;  no  other  law  of  attraction  would  give 
the  general  condition  of  equilibrium  for  all  points  within 
the  spherical  shell. 

An  electrified  pith  ball  remains  at  rest  at  any  point  within  an 
electrified  spherical  shell ;  showing  that  the  law  of  inverse  squares 
is  also  applicable  to  attractions  and  repulsions  between  electrified 
bodies. 

B 


Fig.  46. 

Now,  if  a  body  is  carried  towards  the  center  of  the 
earth,  at  any  point  en  route,  as  at  A  in  Fig.  46,  the  entire 
spherical  shell  of  thickness  AB  can  be  regarded  as  exerting 


FORCE.  71 

no  attraction  whatever  upon  the  body.  The  resultant 
attraction  upon  the  body  is  the  same  as  if  the  earth  were  a 
planet  of  radius  OA.  If  OA  =  Of.  and  the  density  of 
the  earth  were  uniform  at  all  depths,  then  the  mass  of  the 
shaded  portion  would  be  Y%  of  the  total  mass  of  the  earth. 
Hence,  while  the  body  at  A  would  be  only  2000  miles,  (or 
Yt  as  far  as  B}  from  the  earth's  center,  on  the  other  hand  it 
would  be  attracted  by  only  yb  of  the  mass,  and  the  apparent 
weight  would  be  22  X  /^,  or  ^  of  its  weight  at  the  surface. 

That  is,  the  portion  of  the  earth  which  acts  as  an 
attracting  mass  varies  directly  as  OA  \  and  the  effect  of 
distance  inversely  as  OA  ;  hence  the  weight  varies  as 

-.      .    3 

UA    ,  or  directly  as  OA. 


In  example  6,  p.  67,  in  which  a  body  was  weighed  on  different 
planets,  the  weight  of  the  body  varied  directly  as  the  radius  of  the 
planet. 

This  demonstration  assumes  that  the  earth  is  of  uniform 
density  at  all  depths,  but  the  fact  is  that  the  density 
increases  towards  the  center.  Observations  indicate  that 
bodies  carried  downward  exhibit  an  actual  increase  of 
weight  instead  of  a  decrease,  down  to  a  certain  depth, 
beyond  which  the  weight  begins  to  diminish.  It  appears, 
therefore,  that  a  body  really  has  its  maximum  weight,  not 
at  the  surface,  but  a  short  distance  below. 

Mass.  As  stated  in  the  Introduction,  p.  2,  we  have  not 
considered  a  rigid  definition  of  mass,  though  we  have  used 
the  related  ideas  of  weight,  density  and  specific  gravity. 
Our  use  of  this  word,  therefore,  has  been  more  or  less 
tentative.  Our  assumption  has  been  that  every  body  has  in 
it  a  definite  amount  of  matter,  and  we  j  udge  this  amount  by 


72  THEORETICAL   MECHANICS. 

the  weight  of  the  body.  To  judge  the  mass  of  a  body  from 
its  weight  is  safe  enough  where  we  experience  no  greater 
changes  in  the  intensity  of  gravity  than  we  encounter  from 
place  to  place  on  the  earth's  surface.  But  if  we  could  be 
suddenly  transported  to  the  surface  of  the  moon  with  the 
given  body  its  apparent  weightiness  would  be  greatly 
diminished,  though  its  mass  has  remained  unchanged.  In 
fact,  as  we  learned  from  the  law  of  gravitation,  the  mass  of 
a  piece  of  lead  even  would  have  no  weight  at  all  if  it  were 
not  for  the  existence  of  some  other  attracting  mass,  such  as 
the  earth.  But  we  cannot  conceive  of  the  lead  losing  its 
mass  ;  the  very  existence  of  the  body  is  made  known  to  our 
senses  through  its  mass,  or  the  amount  of  material  existing 
in  it.  For  this  reason  mass  is  said  to  be  one  of  the  essential 
properties  of  matter, — essential  to  our  conception  of  its 
existence. 

Masses  are  designated  as  pounds,  ounces,  grams, 
kilograms,  etc., — the  same  as  weights  and  forces.  This 
identity  of  names  as  applied  to  different  things  arises,  of 
course,  from  relations  between  those  things,  but  it  is 
confusing  to  a  beginner  who  has  not  yet  learned  those 
relations.  According  to  the  law  of  gravitation  the  weight 
of  a  body  at  any  given  place,  (that  is,  its  attraction  towards 
the  earth),  is  proportional  to  its  mass.  Expressed 
mathematically,  the  weight  cc  the  mass.  If  we  choose,  we 
may  take  a  lump  of  any  kind  and  call  it  a  unit  mass,  and 
then  we  may  take  as  the  unit  weight,  the  weight  of  this 
unit  mass,  or  of  any  fraction  or  multiple  of  it,  at  any 
standard  place  as  regards  latitude  and  altitude.  A  lump  of 
platinum*  preserved  at  the  Archives  of  Paris  as  a  standard 


*This  is  supposed  to  contain  the  same  mass  as  a  litre  of  water  at  its 
temperature  of  maximum  density,  3° .9  C.,  but  it  has  been  found  that  a  litre  of  pure 
water  at  3°.9  C.  really  weighs  1.000013  Kg. 


MASS.  73 

mass  of  one  kilogram  also  weighs  one  kilogram  at  Paris  ;  it 
weighs  more  than  a  kilogram  at  the  North  Pole,  and  less 
than  a  kilogram  at  the  Equator. 

The  standard  pound  mass  is  a  piece  of  platinum 
preserved  in  the  office  of  the  Exchequer  at  London  ;  the 
weight  of  this  mass  at  London  is  the  standard  pound  weight. 

Density.  The  density  of  a  given  substance  is  the  mass 
of  a  unit  volume  of  that  substance, — expressed  in  grams 
per  c.  c.,  pounds  per  cu.  foot,  etc.  Notice  that  it  is  the 
mass  of  a  unit  volume,  and  not  the  weight.  The  weight  of 
the  body  may  be  different  at  different  places  ;  it  may  even 
be  zero,  as  at  the  center  of  the  earth,  in  which  event  if  the 
density  were  regarded  as  the  weight  of  a  unit  volume,  we 
would  have  to  imagine  the  body  to  exist  without  density. 

Heavy  and  Light  Bodies  Fall  at  the  Same  Rate.     In 

discussing  the  acceleration  of  gravity  we  assumed  that  all 
bodies  fall  at  the  same  rate.  This  is  true  if  we  disregard 
the  frictional  resistance  due  to  the  presence  of  the  air. 
Obviously,  a  light  feathery  body  or  a  piece  of  paper  has 
to  displace  and  drag  itself  along  through  a  quantity  of  air 
that  is  very  large  in  proportion  to  the  weight  of  the  body. 
In  a  vacuum  a  feather  falls  as  fast  as  a  piece  of  metal. 

A  little  thought  will  show  that  it  would  not  be  consistent 
with  our  common  experience  for  a  heavy  body  to  fall  faster 
than  one  half  as  heavy.  If  two  bricks  are  dropped,  one 
from  each  hand,  they  will  reach  the  ground  in  the  same 
time  ;  if  the  two  had  been  glued  together,  there  would  be 
nothing  to  make  the  double  mass  fall  any  faster  than  the 
single  bricks.  In  dynamics  this  would  be  explained  by 
saying  that  while  the  double  mass  has  twice  the  attraction 
downward,  it  also  has  twice  the  mass  to  be  moved  by  that 
attraction. 


CHAPTER  II. 
WORK.        POWER.        ENERGY. 

We  know  that  to  lift  a  weight  or  drag  a  vehicle,  or  to 
overcome  a  resistance  of  any  kind,  requires  the  application 
of  a  force.  If  we  consider  this  applied  force  in  relation  to 
the  distance  through  which  the  body  moves  under  its 
influence,  the  product  of  these  two  values — the  force  and 
the  distance — introduces  one  of  the  most  important  con- 
ceptions in  mechanics.  In  lifting  a  pound  weight  one 
foot,  it  is  said  that  one  Foot-Pound  of  Work  is  done.  To 
lift  a  five  pound  weight  from  the  floor  to  the  top  of  a  table 
three  feet  high,  requires  the  performance  of  15  foot-pounds 
of  work.  If  a  vehicle  requires  a  pull  of  130  pounds  to  drag 
it  along  a  horizontal  surface  the  work  done  for  every  mile 
that  it  is  moved  along  such  a  surface  is  130  X  5280  foot- 
pounds. 

Of  course,  there  may  be  more  than  one  unit  of  work, 
as  a  mile-pound ,  or  a  mile-ton ;  a  kilogram-meter,  or  a  gram- 
centimeter, — according  to  convenience  for  large  and  small 
measurements,  and  to  meet  other  needs.  For  practical  pur- 
poses, however,  it  is  seldom  that  any  unit  of  work  is 
employed  other  than  the  foot-pound  for  the  English  system, 
and  the  kilogram-meter  for  the  metric  system. 

The  work  is  said  to  be  done  by  the  applied  force  and 
against  the  resistance. 

The  time  consumed  is  not  a  factor  in  determining  the 
amount  of  work  done  in  overcoming  a  resistance  through  a 
certain  distance.  The  work  done  in  carrying  a  thousand 
bricks  to  the  top  of  a  given  building  is  the  same  whether  it 


WORK.  75 

is  accomplished  in  a  day  or  a  week.  Work  would  be  done 
faster  in  one  case  than  in  the  other,  but  the  total  number  of 
foot-pounds  is  the  same. 

In  the  sense  of  our  definition,  no  work  is  done  when  a 
weight  is  held  at  rest  in  the  hand,  or  when  it  is  moved 
horizontally.  In  the  first  instance,  it  is  obvious  that  the 
body  is  moved  through  no  distance,  so  that  the  force 
multiplied  by  the  distance  is  zero.  In  the  second  case, 
the  body  is  moved,  but  it  is  neither  raised  nor  lowered. 
The  amount  of  work  done  is  determined  by  the  distance 
through  which  the  resistance  yields  to  the  applied  force. 
With  the  weight  held  in  the  hand  the  muscular  action  and 
the  action  of  gravity  are  both  vertical,  and  as  there  is  no 
vertical  motion  there  is  no  work  done.  The  work  done  in 
carrying  a  weight  up  a  flight  of  stairs  is  the  same  as  if  the 
body  were  lifted  vertically  from  the  lower  floor  to  the  next. 
Work  done  against  gravity  depends  only  upon  difference  of 
level. 

To  make  this  point  sufficiently  clear,  it  seems  almost  necessary  to 
anticipate  a  principle  of  Dynamics.  It  was  stated  on  p.  50  that,  if  it 
were  not  for  the  action  of  gravity,  a  free  and  unobstructed  body 
hurled  in  any  direction  would  move  on  forever  in  a  straight  line  with 
uniform  velocity ;  that  is,  no  force  would  be  required  to  keep  it 
moving,  and  hence  no  work  would  be  done  as  it  moved  mile  after 
mile  through  space.  Even  with  gravity  acting,  if  it  were  not  for  friction, 
it  would  not  require  work  to  keep  a  vehicle  moving  on  a  horizontal 
plane,  once  it  were  started.  On  a  perfectly  smooth  horizontal  plane 
(if  such  could  be  realized),  the  slightest  force  would  start  any  mass 
howsoever  large ;  as  long  as  this  force  continues  to  act,  the  body 
would  keep  on  moving  faster  and  faster ;  when  the  desired  speed  is 
attained  the  force  could  be  withdrawn,  or  cease  to  act,  and  the  body 
would  continue  to  move  on  indefinitely  with  this  velocity.  In  other 
words,  when  a  weight  is  moved  on  a  horizontal  surface  it  is  the 
resistance  of  friction  that  must  be  overcome  and  against  which  work 
is  done ;  the  action  of  gravity  between  the  weight  and  the  earth  is 


76  THEORETICAL   MECHANICS. 

not  overcome,  and  hence  no  work  is  done  against  it.  Indirectly  the 
weight  counts  in  this  way,  that  the  friction  caused  by  such  a  body 
would  be  directly  proportional  to  the  weight,  and  for  that  reason 
more  work  would  be  done  in  moving  a  heavy  body  than  in  moving  a 
lighter  one  through  equal  distances  on  the  same  horizontal  plane. 

Examples: 

1.  A  person   weighing  750  pounds  walks  up  a  flight  of  stairs 
between  two  floors  14  feet  apart.       What  work  is  done  ?      What  if  a 
vertical  ladder  had  been  used?     What  if  he  had  climbed  up  on  a  rope? 

2.  A  horse  drags  a  plow  a  half  mile  against  a  resistance  0/200 
pounds,  as  indicated  on  a  dynamometer.     How  much  work  is  done  ? 

3.  A    person    walking   against   the   wind  has  to  overcome  a 
resistance  of,  say,  4  pounds  per  square  foot.     If  the  surface  meeting 
this  resistance  is  5  square  feet,  what  work  does  he  do  for  every  mile 
walked  ? 

4.  A  gallon  contains  231  cubic  inches.     A  cubic  foot  of  water 
weighs  62.5  pounds.       What  work  will  be  required  to  pump  1000 
gallons  of  water  through  a  vertical  height  of  jo  feet  ? 

5.  How  many  foot-pounds  of  work  will  be  done  in  raising  a 
five-gallon  can  of  alcohol  from  the  floor  to  the  top  of  a  table  jj  inches 
high  ?    (Specific gravity  of  alcohol  -  o.S) 

6.  A  person  weighing  /jj  pounds  rides  up  a  hill  on  a  bicycle 
weighing  2J  pounds.     If  the  hill  rises  70°  from  the  horizontal,  what 
work  is  done  for  every  100  feet  ridden  ? 

7.  In  a  steam  engine  of  o.j  inch  diameter  of  piston  and  12  inch 
stroke,  how  much  work  is  done  during  each  complete  stroke  if  the 
effective  steam  pressure  in  the  cylinder  averages  jo  pounds  per  square 
inch  ?  If  the  engine  is  running  200  revolutions  per  minute,  how  many 
foot-pounds  of  work  are  done  in  a  second  ? 

8.  A  kilogram-meter  is  equivalent  to  how  many  foot-pounds  ? 

Power.  The  horse-power  of  an  engine  is  determined 
by  the  rate  at  which  it  can  do  work.  The  idea  of  Power 
simply  modifies  the  idea  of  Work  by  introducing  a  time 


POWER.  77 

element.  To  raise  a  ton  of  granite  to  the  top  of  a  building 
75  feet  high  requires  150,000  foot-pounds  of  work,  with- 
out regard  to  the  time  consumed.  But  the  engine  that  can 
accomplish  this  amount  of  work  in  one  minute  has  twice 
the  power  of  an  engine  that  would  require  two  minutes  for 
the  same  work. 

A  natural  or  logical  unit  of  Power  would  be  the  ability 
to  do  one  foot-pound  of  work  per  second,  or  some  similar 
value  employing  any  convenient  units  of  force,  distance  and 
time.  It  is  the  practice,  however,  to  use  an  entirely 
arbitrary  unit — the  Horse-Power.  An  engine  or  other 
agency  works  at  the  rate  of  one  H.  P.  if  it  performs  550 
foot-pounds  per  second,  or  33,000  foot-pounds  per  minute. 

The  horse-power  of  an  engine  under  a  given  pressure  of  steam 
could  be  readily  computed  from  its  dimensions  and  speed,  if  the  full 
steam  pressure  acted  throughout  the  entire  length  of  stroke, 
or  even  if  the  average  effective  pressure  were  known.  For  example, 
assume  the  following  conditions:  Diameter  of  piston  9.5  inches; 
length  of  stroke,  12  inches;  number  of  revolutions,  200  per  minute; 
boiler  pressure,  75  pounds  per  square  inch.  If  the  full  steam 
pressure  acted  throughout  the  entire  stroke,  the  work  done  during 
each  double  stroke,  or  revolution,  would  be 

2  X  75  X  7r *~-  X   I2    foot-pounds.* 

4  12 

The  work  done  per  minute  would  be 

2  X  75  X  -      ^—  X  ™-  X  200  foot-pounds, 

and  the  horse-power  supplied  to  the  piston  from  the  energy  of  the 
steam  would  be 

2  X  75  X  *  X  9'5     X  —  X  200  X  — — ,    or  64.4  H.  P. 
4  12 


*Assume  TT  =  3.1416. 


78  THEORETICAL  MECHANICS. 

As  a  matter  of  fact  the  mean  effective  pressure  on  the  piston  is 
considerably  less  than  the  pressure  of  the  steam  as  it  enters  the 
cylinder.  The  inlet  valve  of  an  engine  is  adjusted  in  such  a  manner 
that  the  steam  supply  is  shut  off  when  the  piston  has  completed  only 
a  fraction  of  its  stroke — a  third,  or  a  fourth,  or  at  whatever  point  may 
be  necessary  for  the  best  economy  under  the  given  conditions  of 
steam  pressure  and  load.  After  the  steam  is  cut  off,  the 
remaining  part  of  the  stroke  must  be  completed  by  the  expansion  of 
the  steam  previously  supplied,  and  of  course,  the  pressure  of  this 
steam  diminishes  as  its  volume  (the  space  it  occupies  in  the  cylinder) 
increases. 

It  is  the  function  of  the  governor  to  vary  the  supply  of  steam  as 
needed  to  maintain  a  constant  number  of  revolutions  per  minute. 
There  are  two  types  of  such  governors.  If  the  valve  is  fixed  so  that 
it  always  cuts  off  the  supply  at  the  same  fractional  part  of  a  stroke, 
then  the  governor  regulates  the  steam  supply  by  varying  the  size  of 
an  opening  through  which  the  steam  is  made  to  pass  en  route  to  the 
cylinder.  The  common  Watt  governor  is  of  this  type.  In  an 
automatic  cut-off  engine  the  governor  controls  the  valves  themselves, 
varying  the  point  of  cut-off. 

The  valves  of  an  engine  are  also  adjusted  to  control  the  outlet  of 
steam,  or  the  exhaust.  To  accomplish  easy  running,  enough 
steam  should  be  confined  in  the  idle  end  of  the  cylinder  at  each 
stroke  to  furnish  a  cushion  and  thus  prevent  the  sudden  stopping  and 
consequent  jarring  at  the  end  of  the  stroke.  To  compress  this  steam, 
of  course,  neutralizes  some  of  the  pressure  in  the  active  end  of  the 
cylinder,  so  that  the  mean  effective  pressure  is  still  further  reduced 
from  this  cause. 

In  the  example  selected  for  illustration,  on  the  preceding  page, 
it  was  found  that  the  steam  would  convey  to  the  engine  64.4  H.  P.,  if 
the  total  pressure  of  75  pounds  per  square  inch  were  to  act 
throughout  the  entire  stroke  of  the  piston.  Now,  on  p.  47,  it  was 
stated  that  this  same  engine,  under  exactly  the  same  conditions, 
indicated  24.77  H.  P.,  as  determined  from  an  indicator  card.  This 
shows  that  the  mean  effective  pressure  on  the  piston  was  less  than  the 
boiler  pressure  in  the  ratio  of  -^p  If  ^m  is  the  mean  effective 
pressure,  then 

/>,„  X  ^^5-  X         X  200  X  =  24-77, 


POWER.  79 

or,  Pm  —  28.84  pounds  per  square  inch,  which  will  be  found  to  be 
24.77 
-64^of  75- 

Some  of  the  energy  transferred  to  the  piston  from  the  steam  is 
wasted  by  friction  in  the  engine  itself.  The  rate  at  which  energy  is 
supplied  to  the  piston  is  called  the  Indicated  Horse-Power,  as 
contrasted  with  the  Actual  Horse-Power  available  from  the  engine. 

The  ratio  of  the  Actual  Horse-Power  to  the  Indicated  Horse- 
Power,  expressed  as  a  percentage,  is  the  efficiency  of  the  engine. 

Electrical  power  is  measured  in  Watts,  this  unit  being 
the  rate  at  which  energy  is  conveyed  by  a  current  of  one 
ampere  intensity  under  a  potential  of  one  volt.  One  H.  P. 
in  mechanical  measure  is  equivalent  to  746  Watts. 


Examples : 

1.  How  many  foot-pounds  of  work  can  be  done  in  8  hours  by  a 
50  H.  P.  engine? 

2.  How  many  tons   (2000  pounds)   of  coal  per  hour  can  be 
raised  from  the  ground  to  a  bin  60  feet  above  by  a  10  H.  P.  engine  ? 

3.  What  H.  P.  will  be  required  to  pump  water  to  a  height  of 
120  feet  at  the  rate  of  1000  gallons  per  minute  ? 

Assume  231  cubic  inches  to  a  gallon  and  62.5  pounds  as  the  weight 
of  a  cubic  foot  of  water. 

4.  If  a  horse  can  perform  jjo  foot-pounds  of  work  per  second^ 
at  what  rate  in  miles  per  hour  can  he  drag  a  plow  against  a 
resistance  of  225  pounds  ? 

5.  A  person  carries  a  sign  measuring  3  feet  by  4  feet,  against 
the  wind,  the  resistance  being  6  pounds  per  square  foot.    If  he  walks 
2  miles  per  hour,  at  what  H.  P.  is  he  doing  work  ? 

6.  A  person  weighing  /jo  pounds  runs  up  a  flight  of  stairs  in 
5  seconds.     If  the  stairs  are  2j  feet  long  and  rise  at  an  angle  o 

the  person  is  doing  work  at  the  rate  of  what  H.  P.? 


8o  THEORETICAL  MECHANICS. 

7.  How  many  gallons  of  water  will  a  40  H.  P.  engine  pump  in 
an  hour  from  a  mine  500  feet  deep  ? 

8.  A  house  on  rollers  is  moved  by  means  of  pulleys  and  a  windlass. 
If  the  resistance  to  rolling  is  20  tons,  at  what  rate  can  it  be  moved  by 
a  single  horse  working  at  the  rate  of  one  H.  P.? 

9.  What  must  be  the  H.  P.  of  an  engine  if  it  is  to  be  used  for 
running  a  no  volt  dynamo  that  generates  a  current  of  jo  amperes  f 

10.  What  is  the  H.  P.  of  an  electric  motor  driven  by  a  current 
of  75  amperes  under  potential  of  125  volts  f 

11.  Compute  the  H.  P.  developed  by  an  engine  under  the  following 
conditions:    Diameter  of  piston,  4  inches;  length  of  stroke,  5  inches; 
mean    effective  pressure,    4.2  pounds  per    square  inch;  number  of 
revolutions,  275  per  minute. 

12.  An   engine  has    the  following  dimensions:      Diameter  of 
piston,  12  inches;  length  of  stroke,  18  inches.      At  ijo  revolutions  per 
minute  the  Indicated  H.  P.  is  35.2.      Find  the  mean  effective  pressure. 


Energy.  It  has  been  stated  that  work  is  regarded  as 
being  done  by  the  applied  force  and  against  the  resistance. 
When  a  person  lifts  a  weight  the  muscular  action  is  clearly 
the  applied  force  and  work  is  done  against  the  resistance  of 
gravity.  But  suppose  that  in  this  elevated  position  the 
weight  is  attached  to  a  clock,  or  other  piece  of  mechanism  ; 
by  the  action  of  gravity  the  weight  descends  and  does 
work  against  the  resistance  of  the  machine  driven  by  it.  If 
it  is  a  lo-pound  weight  and  was  lifted  2  feet  above  the  floor, 
the  work  done  in  raising  it  was  20  foot-pounds  ;  this  is 
exactly  equal  to  the  amount  of  work  that  can  be  done  on 
the  machinery  by  the  weight  as  it  returns  to  the  floor.  The 
work  done  in  lifting  the  weight  was  not  wasted;  it  was  simply 
invested  in  the  elevated  mass  as  available  Energy,  ready  to 


ENERGY.  8 1 

be  given  back  in  full  by  performing  20  foot-pounds  of 
work.  A  body  that  is  in  any  way  endowed  with  the  ability 
to  do  work  is  said  to  possess  energy.  If  it  can  do  100 
foot-pounds  of  work  it  possesses  100  foot-pounds  of  energy. 

Potential  Energy.  Hydraulic  elevators  are  operated  by 
storing  water,  either  in  an  elevated  tank  or  in  a  closed  tank 
under  pressure.  Many  machines  are  driven  by  compressed 
air,  the  work  done  in  compressing  the  air  being  given  back 
as  the  molecules  return  to  their  normal  distances  from  each 
other.  The  energy  stored  in  a  clock-spring  by  winding  is 
of  a  sort  much  the  same  as  the  energy  of  compressed  air  ; 
in  all  such  cases  advantage  is  taken  of  the  elasticity  of  the 
material,  which  offers  a  resistance  to  any  force  tending  to 
deform  it  and  thus  stores  up  any  work  done  upon  it.  These 
are  all  examples  of  Potential  Energy,  or  as  it  is  sometimes 
called  Energy  of  Position. 

Two  bodies  that  attract  each  other,  such  as  the  weight 
and  the  earth,  are  endowed  with  this  energy  only  by  the 
act  of  separating  them.  The  energy  invested  in  the  bent 
spring  or  the  compressed  air,  or  in  any  other  elastic  body, 
is  due  to  the  displacement  of  the  particles  from  their  normal 
positions.  It  is  the  tendency  of  the  disturbed  bodies,  or  the 
disturbed  particles  of  the  distorted  body,  to  return  to  their 
normal  positions  that  gives  them  the  ability  to  do  work. 

The  adjective  "Potential"  means  "possible"  and  is 
used  to  signify  that  the  exhibition  of  this  energy  is 
contingent  upon  the  condition  that  the  displaced  body  or 
particles  be  thus  allowed  to  return  to  their  normal  position  or 
positions.  For  instance,  if  the  elevated  weight  is  placed  on 
a  shelf  or  a  table-top  it  can  do  no  work  in  that  position,  but 
must  be  allowed  to  fall  to  the  ground,  if  its  energy  is  to 
become  evident.  The  driving  weight  of  a  clock  is  held  by 


82  THEORETICAL   MECHANICS. 

a  catch  and  performs  its  work  only  as  it  is  released  by  the 
escapement.  The  energy  of  a  bent  spring  and  of  compressed 
air  is  potential  because  of  the  same  contingency. 

The  potential  energy  of  a  body,  as  ordinarily  understood, 
is  not  an  absolute  quantity.  The  potential  energy  of  an 
elevated  weight,  for  example,  is  measured  relatively  to  the 
floor,  or  to  the  ground,  or  some  other  plane  taken  as  a 
standard. 

Kinetic  Energy.  As  contrasted  with  Potential  Energy 
or  Energy  of  Position,  a  body  may  have  Kinetic  Energy 
due  to  its  velocity.  For  illustration,  a  jet  of  water  can  be 
made  to  drive  a  water-wheel  and  thus  perform  work  ;  the 
work  done  by  a  wind-mill  in  pumping  water  is  readily 
traced  to  the  energy  of  the  air  current  ;  it  is  the  kinetic 
energy  of  the  carpenter's  hammer  that  does  the  work  of 
driving  a  nail. 

Energy  of  motion  differs  from  energy  of  position,  also, 
in  this  respect  that  the  former  exists  free  from  all  contin- 
gency. For  that  reason  Kinetic  Energy  is  sometimes  called 
Actual  Energy,  in  distinction  from  Potential  (or  possible) 
Energy. 

The  kinetic  energy  of  a  body  is  the  same,  no  matter 
what  the  direction  of  motion.  It  could  be  measured  experi- 
mentally by  fastening  the  body  in  such  a  way  as  to  compel 
it  to  overcome  a  known  resistance  or  let  it  strike  some 
properly  arranged  obstacle,  and  then  observing  how  far  the 
resistance  yields  before  the  body  comes  to  rest.  The 
simplest  way  to  compute  the  kinetic  energy  of  a  body  from 
its  mass  and  velocity  is  to  find  out  how  far  it  would  rise  // 
it  were  moving  vertically  upward. 

For  example,  suppose  a  given  body  to  have  a  velocity  of 
96.6  feet  per  second,  and  we  wish  to  find  its  kinetic  energy. 


ENERGY.  83 

Now  we  know  from  the  principles  of  acceleration  that  this 
body,  if  it  were  moving  vertically  upward  and  started  from 
the  ground  with  this  velocity,  would  rise  to  a  height  of 
144.9  feet.  In  other  words,  by  virtue  of  its  velocity,  it 
would  be  able  to  lift  itself  to  this  height  against  the  action 
of  gravity.  Its  weight  multiplied  by  this  height  is, 
therefore,  the  amount  of  work  it  can  do,  or  its  energy. 

As  a  further  example,  if  a  bicyclist  wished  to  know  his 
kinetic  energy  at  a  certain  speed,  he  could  start  up  a  hill  of 
known  pitch  with  this  velocity  and,  removing  his  feet  from 
the  pedals,  note  how  far  he  is  carried  up  the  hill.  His 
weight  multiplied  by  the  vertical  height  to  which  he  ascends 
will  represent  his  kinetic  energy  at  the  foot  of  the  hill. 

In  formula  8,  p.  44,  it  was  proved  that  the  distance  a 
body  would  travel  while  gaining  (or  losing)  a  given  velocity 
varies  as  the  square  of  this  velocity.  Hence  the  kinetic 
energy  of  a  body  depends  upon  the  square  of  its  velocity. 
One  body  traveling  5  times  as  fast  as  another  body  of  the 
same  weight  would  have  25  times  as  much  kinetic  energy. 

In  Heat,  Sound,  Light  and  Electricity,  we  find  examples 
of  Molecular  Energy,  as  distinguished  from  Mechanical 
Energy,  which  heretofore  we  have  drawn  upon  exclusively 
for  purposes  of  illustration.  "The  Heat  possessed  by  a 
body  is  explained  as  being  the  Energy  possessed  by  it  in 
virtue  of  the  motion  of  its  particles.  Just  as  a  swarm  of 
insects  may  remain  nearly  at  the  same  spot  while  each 
individual  insect  is  energetically  bustling  about,  so  a  warm 
body  is  conceived  as  an  aggregation  of  particles  which  are 
in  active  motion  while  the  mass  as  a  whole  has  no  motion."1 
Such  a  body  has  invisible  Molecular  Kinetic  Energy. 
Moving  en  masse,  with  visible  motion,  its  energy  would  be 


^Darnell's  Text  Book  of  the  Principles  of  Physics,  p.  48. 


84  THEORETICAL  MECHANICS. 

called  Mechanical.  Obviously,  it  may  have  both  at  the 
same  time.  Or,  if  raised  to  an  elevated  position,  its 
mechanical  energy  would  be  potential, — its  molecular 
of  course,  remaining  unchanged. 

The  Chemical  Energy  of  gunpowder  is  a  familiar 
example  of  Potential  Molecular  Energy, — though  perhaps  it 
would  better  be  called  Atomic  rather  than  Molecular.  The 
different  constituents  of  gunpowder — sulphur,  saltpetre  and 
charcoal — have  a  chemical  affinity  for  each  other,  by  virtue 
of  which  they  tend  to  come  together  and  form  new 
compounds,  just  as  a  weight  tends  to  fall  to  the  earth.  In 
the  gunpowder  the  constituents,  while  intimately  mixed 
together,  are  still  separated  from  each  other  chemically,  and 
hence  they  have  potential  energy.  Once  the  impulse  is 
given,  like  releasing  the  elevated  weight  from  the  shelf, 
these  chemical  constituents  rush  together,  atom  clashing 
against  atom,  forming  new  gaseous  compounds,  and 
generating  an  enormous  amount  of  heat.  Some  of  the 
original  chemical  energy  (molecular  potential)  is  thus 
converted  into  Heat  (molecular  kinetic). 

Any  form  of  molecular  energy  is  measurable  in 
mechanical  units. 

The  Watt,  as  already,  stated,  is  equivalent  to  rU  H.  P. 
This  is  really  a  conversion  of  Electrical  into  mechanical 
power,  and  not  a  conversion  of  energy.  The  mechanical 
equivalent  of  an  electrical  unit  of  energy  could  easily  be 
derived,  but  is  not  needed  in  dealing  with  currents  of 
electricity,  because  a  current  always  involves  a  time  element, 
which  we  also  found  to  be  involved  in  power  but  not  in 
work. 

The  unit  of  Heat  is  the  amount  of  heat  necessary  to 
raise  a  unit  weight  of  water  one  degree  in  temperature. 
Taking  a  pound  as  the  unit  of  weight  and  using  the 


ENERGY.  85 

Fahrenheit  thermometric  scale,  the  heat  necessary  to  raise  a 
pound  of  water  i°  F.  is  equivalent  to  778.5  foot-pounds  of 
work. 


Transference  and  Transformations  of  Energy.  Any  one 
of  the  different  kinds  of  energy  can  be  transformed,  directly 
or  indirectly,  into  any  other  kind.  Mechanical  energy  is 
readily  changed  from  the  kinetic  form  to  the  potential,  and 
vice  versa.  Mechanical  energy  of  either  form  can  be  con- 
verted into  heat,  sound,  light  or  electricity.  And  any  one  of 
these  various  molecular  forms  can  be  transformed  into  any 
other,  or  into  mechanical  energy.  Chemical  energy  is 
likewise  convertible,  especially  into  heat  and  electricity. 

Energy  can  also  be  transferred  from  one  body  to  another 
and  thus  transmitted  from  place  to  place.  Whenever  work 
is  done  there  is  a  transformation  or  a  transference  of  energy, 
or  both.  In  fact,  at  this  point  our  definition  of  work  might 
well  be  revised  in  accordance  with  this  larger  view. 

But  with  all  these  changes  of  energy  from  body  to  body 
and  from  one  form  to  another,  there  is  no  actual  gain  or 
loss  of  energy.  It  is  always  a  case  of  adding  and  subtracting 
the  same  quantity.  The  work  done  by  one  body  exerting 
a  force  on  a  second  body  is  j  ust  equal  to  the  work  done  upon 
this  resisting  body,  or  in  other  words  the  first  body  yields 
up  to  the  second  body  an  amount  of  energy  equal  to  the 
number  of  foot-pounds  of  work  done.  So  far  as  we  know 
the  total  energy  of  the  universe  is  an  unchanging  quantity — 
a  definite  number  of  foot-pounds.  This  hypothesis  is  known 
as  the  Conservation  of  Energy,  and  is  accepted  as  the  most 
fundamental  principle  of  physics. 

A  simple  illustration  is  in  the  conversion  of  mechanical  energy 
from  the  kinetic  form  to  the  potential,  and  vice  versa.  Suppose  that 


86  THEORETICAL  MECHANICS. 

a  ten  pound  weight  is  projected  vertically  upward  with  a  velocity  of 
96.6  feet  per  second.  At  the  instant  of  leaving  the  ground  it  has 
10  X  144.9  or  1449  foot-pounds  of  kinetic  energy,  because  by  virtue 
of  its  velocity  of  projection  it  is  capable  of  rising  to  a  height  of  144.9 
feet  against  a  gravitational  resistance  of  10  pounds.  But  when  it  has 
reached  the  highest  point,  ready  to  start  downward — what  then  ? 
In  this  position  it  has  1449  foot-pounds  of  potential  energy.  And  if 
at  that  instant  the  weight  could  be  caught  on  a  hook  or  a  shelf,  it 
could  then  be  attached  to  a  clock  or  other  mechanism  and  its 
potential  energy  used  for  any  purpose  desired, — in  which  event  the 
body  would  return  to  the  ground  leisurely  and  perhaps  with  a  velocity 
hardly  perceptible.  But  if  it  is  not  thus  restrained,  but  is  allowed  to 
fall  freely  from  this  height  of  144.9  feet>  it  will  fall  in  the  usual 
manner — at  an  accelerated  rate.  On  the  downward  trip  it  will  gain 
velocity  as  fast  as  it  lost  it  on  the  upward  trip,  and  when  it  reaches 
the  ground  it  will  have  the  same  kinetic  energy  that  it  started  with. 
Instantly  it  strikes  the  ground,  however,  this  mechanical  energy  of 
motion  is  converted  into  sound,  heat,  and  perhaps  light  and 
electricity.  It  follows,  therefore,  that  when  a  body  is  projected 
upward  its  kinetic  energy  is  gradually  converted  into  potential 
energy,  and  back  again  to  the  kinetic  form  as  it  descends.  At  the 
summit  it  has  only  potential  energy.  For  every  foot  that  it  rises,  if 
its  weight  is  10  pounds,  it  gains  10  foot-pounds  of  potential  energy 
and  loses  an  equal  amount  of  kinetic  energy.  Half  way  up  half  its 
energy  is  potential  and  half  kinetic.  The  total  energy,  however, 
remains  the  same. 

The  generation  of  an  electric  current  affords  another  good 
illustration  of  transformation.  The  current  itself  is,  of  course,  an 
example  of  transmission  or  transference  of  energy.  If  the  current 
is  generated  by  a  battery — a  voltaic  current — it  is  a  case  of  direct 
conversion  of  chemical  into  electrical  energy.  If  it  is  furnished  by  a 
dynamo — a  magnetic-electric  current — its  energy  can  be  traced  back 
step  by  step,  first  to  the  mechanical  energy  of  the  machinery,  and 
thence  to  the  energy  of  the  steam  generated  by  the  heat  of  the 
furnace.  The  heat  itself  originated  in  chemical  energy  of  the  coal 
and  oxygen  uniting  with  each  other  to  form  carbonic  acid  and  the 
other  gases.  We  might  go  a  step  further  and  say  that  the  energy  of  the 
coal  and  oxygen  came  from  the  rays  of  the  Sun  which  in  some  past  age 
decomposed  a  lot  of  carbonic  acid  gas,  giving  the  carbon  to  make  the 
wood  of  a  tree  and  liberating  the  oxygen  to  form  part  of  the  atmosphere. 


ENERGY.  87 

This  dynamo  might  have  been  driven  by  a  water  wheel,  power 
being  furnished  by  a  jet  of  water.  And  perhaps  the  water  was  first 
stored  for  some  length  of  time  in  a  reservoir.  Looking  still  further 
we  find  that  it  was  raised  to  the  reservoir  from  the  ocean  at  the 
expense  of  energy  of  the  Sun. 

There  is  hardly  an  instance  in  which  the  power  used  by  man 
could  not  be  traced  to  the  Sun.  The  energy  radiated  from  the  Sun 
itself  may  be  due,  all  or  in  part,  to  chemical  action  now  going  on 
there  or  to  radiation  from  a  molten  mass,  or  to  still  other  causes 
equally  probable  and  based  upon  even  more  recent  theories. 

Examples : 

1.  What  is  the  potential  energy  of  a  body  weighing  zoo  kg.  at 
an  elevation  of  22  meters  ? 

2.  A  body  weighing  5  pounds  is  dropped  from  a  height  of  1000 
feet.     What  kind  of  energy  and  how  much  did  it  have  when  it  started  ? 

At  the  end  of  one  second  what  change  will  have  taken  place  ?  At  the 
end  of  two  seconds  f  At  the  end  of  three  seconds  ?  fust  as  it  reaches 
the  ground  f  After  it  has  struck  the  ground  ? 

3.  A  body  is  projected  vertically  upward  with  a  velocity  of  322 
feet  per  second.      How  much  energy  has  it  and  of  what  kind?     What 
change  will  have  taken  place  at  the  end  of  one  second?    At  the  end  of 
two  seconds  ?   At  the  end  of  three  seconds  ?    At  the  end  of  ten  seconds  ? 

4.  What    transformations    of    energy    take  place  during   the 
.oscillations  of  a  pendulum  f 

5.  A   bicyclist  weighing  155  pounds  mounted  on  a  2j-pound 
wheel  rides  at  a  velocity  of  20  miles  an  hour  on  a  horizontal  plane. 

(a)  What  is  the  combined  kinetic  energy  of  the  man  and 
wheel? 

(b]  If  he  comes  to  a  slope  rising  at  an  angle  of  12°  and 
removes  his  feet  from  the  pedals,  how  far  will  he  progress  up 
the  hill  f 


88  THEORETICAL  MECHANICS. 

6.  The  same  person  riding  at  the  same  rate  comes  to  a  hill  of 
unknown  pitch,  and  finds  that  his  velocity  is  sufficient  to  carry  him 
75  feet  UP  the  slope.      Find  the  pitch  of  the  hill  from  the  horizontal. 

7.  In  the    Yosemite    Water-Fall  the    total  drop  is  about  jooo 
feet. 

(a]  What  transformations  of  energy  occur  ? 

(b]  When   the   water  'strikes  at  the  foot  of  the  Falls  its 
mechanical  kinetic  energy  is  converted  mainly  into  heat.     If  there 
were  no  loss  from  friction,  etc.,  during  the  drop,  and  if  all  the 
energy  were  converted  into  heat  at  the  instant  of  striking,  how 
much  would  the  temperature  of  the  water  be  raised? 

8.  A  body  has  a  velocity  of  100  feet  per  second  and  weighs  21 
Pounds.     What  kinetic  energy  has  it  f 

9.  A  body  weighing  50  pounds  and  having  a  velocity  of  40  feet 
per    second  moves    along  a  horizontal  plane    against  a  fractional 
resistance  of  2  pounds. 

(a)  How  far  will  it  travel  before  coming  to  rest? 

(b]  How  long  before  it  will  come  to  rest  f 

HINT:  First  compute  the  energy  of  the  body  by  finding  out 
how  far  it  would  rise  against  its  own  weight  if  it  were  moving 
vertically  upward. 

10.  A  freight  car  weighing  jo  tons  is  moving  on  a  horizontal 
track  at  the  rate  of  40  miles  per  hour.     If   the  total  resistance  occa- 
sioned by  the  brakes  and  friction  on  the  tracks  is  2  tons,  how  far  will 
the  car  move  before  it  is  brought  to  rest? 

11.  What  transformations  of  energy  take  place  when  a  fire- 
cracker explodes1?     When  a  steam-whistle  blows?     When  a  gong  is 
sounded  ? 

12.  An  electric  current  is  used  to  drive  a  motor  attached  to  a 
lathe.     What  transformation  of  energy  takes  place  ?     What  if  the 
current  had  been  used  for  an  incandescent  lamp?    For  electro-plating? 
For  ringing  an  electric  bell  f    For  electric  welding  ? 


WORK. 


89 


Graphical  Representation  of  Work.  Using  two  rectangular 
axes  in  the  manner  explained  on  pp.  46  and  47,  Work  can  be 
represented  as  an  area — a  rectangle,  of  which  one  side  or  dimension 
stands  for  either  the  applied  force  or  the  resistance  and  the  other 
dimension  stands  for  the  distance  through  which  the  resistance  is 
displaced.  For  example,  to  represent  the  work  done  in  raising  a 
weight  of  600  pounds  to  a  height  of  50  feet,  let  us  assume  a  vertical 
scale  of  i  inch  -  200  pounds  and  a  horizontal  scale  of  i  inch  =  10  feet. 
From  the  origin  O,  Fig.  47,  measure  off  on  the  F-axis  the  length 
OA  =  3  inches  (to  represent  the  weight,  600  pounds),  and  on  the 
Jf-axis  measure  off  the  distance  OB  =  5  inches  (to  represent  the 
displacement  of  50  feet).*  Since  OA  represents  a  force  and  OB  a 


Fig.  47- 

displacement  caused  by  this  force,  it  follows  that  the  work  done  is 
represented  in  the  diagram  by  the  area  of  the  rectangle  OB  A  C,  or 
what  is  the  same  thing  OA  X  OB. 

If  the  diagram  is  analyzed  still  further  it  will  be  seen  that  this 
conclusion  is  an  entirely  consistent  one.  Since  the  weight  remains 
constant  throughout  the  displacement,  a  perpendicular  erected  at  B 
or  at  any  point  between  O  and  B,  and  equal  to  OA,  will  represent 
the  magnitude  of  the  force  for  that  particular  point  in  the 
displacement.  The  locus  of  the  extremities  of  such  perpendiculars 
will  be  a  line  through  point  A  parallel  to  OB. 

If  the  force  had  been  variable  the  locus  A  C  would  be  an  irregular 
line  as  yy±  in  Fig.  36,  p.  49.  The  work  done,  however,  would  still  be 
the  area  included  between  this  line  and  the  two  axes,  or  O  xy-^y  in 
Fig.  36.  In  fact,  this  general  method  of  interpreting  an  area  is  the 


*The  printed  diagram  is  reduced  to  one  quarter  of  this  scale. 


9o 


THEORETICAL   MECHANICS. 


same  as  was  explained  on  pp.  48  and  49  in  connection  with  accelerated 
motion,  except,  of  course,  that  the  co-ordinates  in  that  case  were  made 
to  represent  entirely  different  quantities  from  those  we  are  now 
dealing  with  in  connection  with  Work. 

Between  these  two  extreme  cases  of  a  uniform  force  and  an  irregu- 
larly changeable  one,  there  are  circumstances  under  which  the  force 
changes  uniformly,  and  for  which  the  diagram  yields  a  triangular 
area  to  represent  the  work  done.  This  is  always  the  case  when  a 
force  is  applied  so  as  to  gradually  elongate  a  wire  or  rod,  or  a  spiral 
spring,  and  in  all  other  cases  where  a  stress  is  produced  in  a 
body — as  in  bending  a  beam,  or  twisting  a  shaft,  or  compressing  a 
block  of  any  kind.  Imagine  a  small  load,  say  a  pound  weight, 
applied  to  a  spiral  spring;  the  spring  elongates  a  certain  amount  and 
stops,  showing  that  its  ability  to  resist  the  elongating  force  gradually 
increases  as  it  is  stretched.  If  the  load  be  doubled  by  adding  a 


second  weight  the  spring  will  stretch  still  further, — until  the  internal 
resistance  again  becomes  equal  to  this  external  load.  Now,  if  we 
measure  the  length  of  the  wire  under  this  load  we  will  find  that  the 
elongation  for  the  two  pounds  has  been  just  twice  as  great  as  for  one. 
If  we  add  a  third  pound  the  elongation  becomes  three  times  as  great  t 
the  internal  resistance,  of  course,  increasing  in  the  same  ratio  in 
order  to  balance  the  weight.  In  other  words,  the  internal  resistance 
of  the  spring  is  directly  proportional  to  the  elongation.  The  same 
idea  applies  to  all  bodies,  whatever  the  nature  of  the  stress.  This  is 
called  Hooke's  Law  which  asserts  that  within  certain  limits  the 
internal  resistance  of  a  distorted  body  is  directly  proportional  to  the 
amount  of  distortion  or  deformation.  When  the  internal  resistance 


WORK.  QI 

becomes  equal  to  the  external  force  the  distortion  ceases;  and  hence 
for  bodies  at  equilibrium  under  stress — the  usual  case — the  law  might 
be  expressed  in  the  converse  form,  viz.,  that  the  distortion  is 
directly  proportional  to  the  distorting  force. 

In  graphical  form  the  work  done  in  elongating  the  spring  by 
adding  successive  weights  of  one  pound  each  would  be  represented  as 
in  Fig.  48.  Let  OA'  represent  the  force  of  one  pound  and  OB'  the 
elongation  produced  by  this  force;  the  work  done  thereby  will  be  the 
area  OB'C'A'.  Then  a  second  weight  was  added,  making  two  pounds 
in  all,  as  B'A",  and  producing  an  additional  elongation  B'B",  or  a 
total  of  OB"  equal  to  2  OB'.  As  we  go  on  stretching  the  spring 
farther  and  farther,  the  area  representing  the  work  increases  step  by 
step  in  the  manner  pictured  in  the  diagram.  The  work  done  in 


Fig.  49- 


stretching  the  spring  through  the  distance  represented  by  B"'B""  is 
four  times  as  great  as  in  stretching  it  through  the  same  distance  OB' 
at  the  first  stage  of  the  elongation. 

If  the  tension  had  been  applied  to  the  spring  more  gradually, 
increasing  continuously  instead  of  a  pound  at  a  time,  the  diagram 
would  have  been  as  in  Fig.  49.  If  the  tension  increases  uniformly 
from  zero  to  BC,  while  the  spring  is  elongated  an  amount  represented 
by  OB,  the  work  done  will  be  represented  by  the  area  of  the 
triangle  OBC, — which  is  only  half  as  great  as  if  the  maximum  load 
BC  acted  throughout  the  entire  distance,  as  in  Fig.  47.  Or,  the  average 
load  is  only  a  half  of  BC. 

What  is  true  in  this  respect  for  the  elongation  of  a  spring  is  true 
for  a  body  distorted  in  any  manner. 


92  THEORETICAL   MECHANICS. 

In  the  Indicator  Diagram  shown  in  Fig.  34,  p.  47,  each  half  of 
the  diagram  shows  the  changes  of  steam  pressure  in  one  end  of  the 
cylinder  for  a  complete  or  double  stroke  of  the  piston.  The  area 
enclosed  within  ABCDEA,  Fig.  50,  represents  the  work  done  in  one 
end  of  the  piston,  and  the  corresponding  area  A'B'C'D'E'A  is  the 
work  done  in  the  other  end.  The  total  work  done  by  the  steam  for  a 
complete  stroke  is  the  sum  of  the  two.  In  the  diagram  the  two 
areas  overlap  in  part,  but  this  is  only  a  matter  of  convenience  in  the 
mechanical  process  of  taking  the  card  from  the  engine;  in  computing 
the  work  done  this  area  is  counted  twice. 

When  the  piston  is  at  the  right  end  of  a  stroke  in  the  diagram,  a 
small  amount  of  steam  is  confined  in  that  end  of  the  cylinder  under 
pressure  D'A  to  form  a  cushion,  as  explained  on  p.  78,  while  the 
other  end  at  that  instant  is  in  communication  with  the  air  and  hence 


D 


Fig.  50. 


has  no  pressure  above  the  atmospheric.  In  this  position,  steam  is 
admitted  to  the  right  end  and  the  pressure  in  that  end  instantly 
jumps  from  D'A  to  the  boiler  pressure  D'B.  The  piston  moves  to  the 
left.  The  right  port  apparently  remains  open  to  the  full  boiler 
pressure  for  about  one-tenth  of  the  stroke,  when  it  closes  (at  point  C)  • 
for  the  balance  of  the  stroke  the  steam  works  by  expansion, 
gradually  falling  off  from  the  full  pressure  at  C  to  the  atmospheric 
pressure  at  D.  At  D  the  piston  completes  the  half  stroke  and  starts 
back.  The  right  end  of  the  cylinder  is  now  open  to  the  atmosphere 
and  remains  so  until  point  E  is  reached,  when  it  closes  for  the 
purpose  of  retaining  sufficient  steam  to  form  the  cushion  by  being 


WORK.  93 

compressed  from  £  to  A.  If  it  were  not  for  this  compression  the 
work  done  by  this  end  of  the  piston  for  a  complete  stroke  would  be 
represented  by  the  somewhat  triangular  area  DD'BCD.  Out  of  this 
area  we  must  subtract  the  area  ED' A,  lost  by  compression  during  the 
return  stroke. 

While  this  subtraction  is  mathematically  correct  it  does  not  give 
a  correct  idea  of  the  actual  transaction.  The  work  of  compression 
represented  by  the  area  ED' A  is  really  done  "by  the  steam  in  the  other 
end  of  the  cylinder,  and  properly  should  be  subtracted  from  the  area 
D'DB'C'D'.  The  two  halves  of  the  diagram  should  be  read 
together.  The  right  end  actually  does  the  full  amount  of  work 
represented  by  the  area  DD'BC,  a  part  of  this  being  used  up  in  the 
work  of  compressing  the  hold-over  steam  in  the  other  end  of  the 
cylinder  (area  E'DA'}  and  the  balance  being  given  to  the  engine  for 
useful  work.  We  must  not  only  read  the  conditions  in  the  two  ends 
of  the  cylinder  at  the  same  time,  but  also  in  their  relations,  each  to 
the  other.  As  the  pressure  in  the  right  end  of  the  cylinder  falls  from 
B  to  D,  the  pressure  in  the  other  end  changes  from  D'  to  B ',  the 
compression  commencing  at  E'.  From  the  intersection,  X,  of  these 
two  lines  the  pressure  on  the  driving  end  of  the  piston  is  actually  less 
than  on  the  other  end,  and  if  it  were  not  for  the  kinetic  energy  of  the 
fly-wheel  the  piston  would  not  complete  its  stroke  but  would  bound, 
back  before  reaching  D. 

Hence  the  area  DD'BCD  minus  area  E'DA'  is  the  effective  work 
done  during  a  half  revolution  of  the  engine,  and  D'DB'C'D'  minus 
ED  A  is  the  effective  work  for  the  other  half.  Putting  this  in 
algebraic  form  we  have 

(DD'BCD  -  E'DA'}  -\-  (D'DB'C'D1  -  ED1  A} 

as  the  total  work  done  during  a  complete  stroke  or  revolution.  This 
can  be  transformed  into 

(DD'BCD  -  ED' A}  +  (D'DB'C'D'  -  E'DA'),  or 

ABCDEA  +  A'B'C'D'E'A'-, 

thus  proving  what  was  before  stated,  that  this  last  result  represents 
the  actual  mathematical  value  of  the  work  done,  even  if  it  does  not 
picture  the  transaction  correctly. 

The  Mean  Effective  Pressure,  referred  to  on  p.  76,  would  be  the 
average  of  an  infinite  number  of  lines  drawn  like/^  in  the  diagram. 
It  is  the  combined  areas  ABCDEA  +  A'B'C'D'E'A'  divided  by  the 
length  of  stroke,  DD'. 


CHAPTER  III. 
CENTER  OF  GRAVITY. 

Center  of  Figure.  The  Center  of  Figure  of  a  straight 
line  is  at  its  middle  point  ;  as  much  of  the  figure  lies  on  one 
side  of  the  point  as  on  the  'other.  With  equal  facility  we 
can  locate  the  center  of  figure  of  a  circle,  an  ellipse,  a 
parallelogram,  a  sphere,  a  cylinder,  or  any  other  sym- 
metrical figure.  It  coincides  with  the  center  of  symmetry, 
as  defined  in  geometry.  The  center  of  figure  of  a  spherical 
shell  is  the  same  as  if  it  were  a  solid  sphere  ;  a  hollow  box 
has  the  same  center  as  a  solid  block  of  the  same  size  and 


shape  ;  likewise  the  center  of  a  length  of  pipe  is  not  in  the 
material  of  the  pipe  but  at  the  middle  point  of  the  axis  of 
the  enclosed  cylindrical  space. 

Even  if  the  figure  is  not  strictly  symmetrical,  if  it 
possesses  some  degree  of  regularity,  there  may  be  in  it  a 
point  which  may  properly  be  called  its  center  of  figure.  A 
triangle  can  be  imagined  to  be  made  up  of  an  infinite  number 
of  parallel  straight  lines,  as  in  Fig.  51  and  the  center  ot 


CENTER   OF   GRAVITY.  95 

each  line  will  be  in  the  median  CD.  Since  all  these  lines 
taken  together  constitute  the  triangle,  the  center  of  the 
triangle  must  be  somewhere  in  the  median  CD,  If  a 
second  set  of  lines  were  drawn  parallel  to  AC,  or  BC, 
their  centers  would  all  lie  in  a  second  median.  The 
intersection  of  two  of  its  medians  is  therefore  the  center 
of  figure  of  the  triangle. 


Center  of  Mass.  If  a  geometrically  symmetrical  body  is 
also  composed  of  uniformly  distributed  particles,  the  center 
of  figure  is  the  position  of  a  central  particle.  And  if, 
furthermore,  the  body  is  of  a  strictly  homogenous  substance 
or  of  uniform  density,  the  central  particle  is  also  an  average 
point  about  which  the  mass  of  the  body  is  distributed, — or 
the  center  of  mass,  as  it  is  commonly  called. 

If  this  body  howsoever  symmetrical  in  shape  is  not  of 
uniform  density  the  center  of  mass  is  not  so  readily  located. 

If  a  piece  of  lead  is  glued  to  the  end  of  a  cork,  the 
central  particle  of  the  combined  mass  may  or  may  not  be  at 
the  center  of  figure,  but  the  center  of  mass  is  evidently 
nearer  the  lead. 


Center  of  Gravity.  According  to  the  law  of  gravitation, 
every  particle  of  a  body  participates  in  the  action  of  gravity, 
and,  (if  the  body  is  small  in  comparison  with  the  size  of  the 
earth,  so  that  the  lines  from  its  different  particles  to  the 
center  of  the  earth  may  be  regarded  as  parallel) ,  the  weight 
of  the  body  is  the  sum-total  of  these  minute  attractions.  For 
many  purposes  in  mechanics,  this  downward  tendency  or 
weight  of  the  body  may  be  represented  as  a  single  force, 
acting  at  an  average  or  central  point  called  the  Center  of 
Gravity. 


96  THEORETICAL  MECHANICS. 

The  importance  of  this  conception  is  that  the  action  of 
gravity,  or  any  other  force  acting  equally  and  in  parallel 
directions  on  all  the  particles  of  the  body,  is  the  same  in  effect 
as  'if  the  entire  mass  of  the  body  were  condensed  in  its 
center  of  gravity,  the  remaining  particles  of  the  body  being 
imagined  as  weightless. 

Since  the  laws  of  mechanics  deal  with  masses  only 
through  their  relation  to  forces  acting  upon  them,  there  is 
no  discrimination  that  we  shall  need  to  exercise  in  using  the 
terms  "center  of  mass,"  ''center  of  gravity"  and  "center  of 
weight;"  we  may  use  one  for  another  without  danger  of 
error. 

To  Locate  the  Center  of  Gravity  of  a  Body.  It  has  been 
shown  that  the  center  of  figure  of  any  symmetrical  body 
can  be  found  by  simple  geometric  construction,  and  also 
that  if  its  density  is  uniform  its  center  of  mass  and  center 
of  figure  will  coincide.  If  it  is  a  hollow  body— a  tube,  box, 
ring  or  spherical  shell — this  central  point  is  still  called  the 
center  of  mass,  even  though  there  is  no  central  particle  of 
the  substance. 

Having  located  the  center  of  mass  of  each  of  several 
symmetrical  bodies  by  geometric  methods,  we  can  use  the 
results  to  prove  the  following 

Proposition: — If  a  body  is  supported  freely  and  loosely  on 
a  pivot  it  will  adjust  itself  so  that  the  center  of  gravity  will 
be  vertically  below  (or  above)  the  point  of  support. 

(a)  Prepare  two  pieces  of  cardboard,  one  triangular  and  the 
other  a  parallelogram.  Locate  the  center  of  each  by  geometry.  In 
each  make  two  pin  holes  near  the  edges  and  not  too  near  each  other. 
Prepare  a  plumb-line  by  tieing  a  screw,  or  other  small  object,  to  a 
fine  thread.  On  a  fine  needle  driven  into  the  wall  suspend  the 
triangle  from  one  of.  the  pin  holes,  and  hang  the  plumb-line  from  the 


CENTER   OF   GRAVITY.  97 

same  support.  When  both  have  come  to  rest,  mark  the  lower  edge 
of  the  triangle  where  it  crosses  the  plumb-line.  Remove  from  the 
pivot  and  draw  a  line  from  the  point  of  support  to  the  point  marked 
on  the  edge.  Now  suspend  the  triangle  from  the  second  pin  hole, 
and  determine  a  second  line.  Do  these  lines  intersect  at  the  center 
of  figure  as  determined  by  construction  ?  If  not,  determine  the 
amount  of  error  by  measuring  the  distance  between  the  two  locations. 
When  the  triangle  was  supported  on  the  needle  what  was  the  position 
of  the  center  of  gravity  relatively  to  the  point  of  support  ? 

In  the  same  manner  hang  the  parallelogram  on  the  needle  from 
each  of  the  pin  holes  in  succession,  and  note  results. 

(b]  Remove  the  cover  and  bottom  of  a  pasteboard  box.  Stick 
a  pin  in  one  of  the  edges  so  that  the  pin  will  run  full  length  into  the 
cardboard  parallel  to  the  side  of  the  box.  Tie  the  plumb-line  at 
about  its  middle  point  around  the  pin  just  at  its  head.  Fasten  the  top 
of  the  plumb-line  to  some  convenient  support  and  allow  the  bob  to 
drop  through  the  hollow  of  the  box.  When  both  have  come  to  rest 
mark  the  lower  edge  of  the  box  where  it  crosses  the  plumb-line. 
After  removing  the  plumb-line  measure  the  distance  to  the  nearest 
corner  from  the  pin  hole  and  also  from  the  marked  point  where  the 
plumb-line  crossed.  Are  these  points  located  symmetrically? 

By  means  of  a  needle  and  thread  connect  the  middle  points  of  the 
opposite  sides  of  the  box,  thus  locating  the  center  of  the  box.  Again 
suspend  as  before  and  see  if  the  plumb-line  passes  through  the 
center  of  figure. 

These  experiments  will  suffice  to  show  that  the  center  of 
gravity  of  a  body  ordinarily  assumes  a  position  vertically 
below  the  point  of  support.  This  is  a  simple  fact  that 
hardly  needed  demonstration  ;  it  is  in  perfect  accord  with 
an  endless  number  of  phenomena  that  we  observe  from  day 
to  day. 

To  support  a  body  from  a  single  point  with  its  center  of 
gravity  vertically  above  the  point  of  support — such  as  a  cone 
balanced  on  its  apex,  or  a  parallelepiped  on  an  edge — is 
theoretically  possible,  but  in  practice  the  most  skillful 
equilibrist  would  require  some  small  area — more  than  a 
mathematical  line  or  point — for  a  base  of  support. 


98 


THEORETICAL   MECHANICS. 


Equilibrium,  (a)  When  a  body  is  supported  from  a 
pivot,  with  its  center  of  gravity  vertically  above  or  below 
the  point  of  support,  it  is  said  to  be  in  a  position  of 
equilibrium.  The  weight  of  the  body,  acting  like  a  single 
downward  force  exerted  at  the  center  of  gravity,  is  balanced 
or  equilibrated  by  the  resistance  of  the  pivot.  If  the 

center  of  gravity  were  not  in 
vertical  line  with  the  point  of 
support,  as  in  Fig.  52,  the  pivot 
could  not  furnish  a  resistance 
equal  and  opposite  to  the  force 
exerted  at  C  by  the  weight  of 
the  body.  To  assume  a  position 
of  equilibrium  the  body  must 
rotate  around  the  pivot  until  the 
vertical  CD  passes  through  P. 

(b)  Problems  of  equilibrium  are  more  frequently  met 
with  in  bodies  resting  on  a  base  of  support;  pivoted  bodies 
came  first  in  order  of  consideration,  not  because  of  their 


greater  importance,  but  because  it  is  simpler  to  deal  with  a 
single  point  of  support  than  with  an  area  of  support.  If  a 
rectangular  block  rests  on  one  of  its  faces  on  a  horizontal 
plane  (Fig.  53-^),  it  is  in  equilibrium,  because  the  vertical 


CENTER   OF   GRAVITY.  99 

drawn  through  the  center  of  gravity  passes  through  the 
base  of  support,  and  the  force  of  gravity  is  balanced  by  the 
resistance  of  the  object  supporting  the  base.  If  the  block 
were  not  rectangular,  as  in  Figs.  53-^,  c,  d  and  e,  it  would 
be  in  equilibrium  or  not,  depending  upon  the  area  of  base, 
the  vertical  height  and  the  acuteness  of  the  angles.  Figs. 
b  and  c  have  the  same  area  of  base  and  the  same  vertical 
height,  but  the  angles  of  c  are  such  that  the  vertical  from 
the  center  of  gravity  falls  outside  the  base  of  support  and 
the  body  is  not  in  equilibrium.  Figs,  b  and  d  have  the  same 
angles  and  the  same  vertical  height,  but  Fig.  d  is  not  in 
equilibrium  because  of  its  smaller  base.  Figs,  c  and  d  would 
both  be  toppled  over  by  their  own  weight. 


A  body  at  equilibrium  on  a  horizontal  plane  might  not 
be  at  equilibrium  if  the  plane  were  inclined,  as  shown  in 
Fig.  54.  A  sphere  is  always  at  equilibrium  on  a  horizontal 
plane,  but  is  never  so  on  an  inclined  plane. 

If  a  body  rests  on  several  isolated  points,  as  a  surveyor's 
tripod,  or  a  table,  the  base  of  support  is  the  convex  polygon 
that  would  be  determined  by  winding  a  thread  around  the 
external  points  of  support.  There  may  be  many  other 
supporting  points  within  the  perimeter  thus  determined, 
but  as  they  do  not  extend  the  area  of  support  they  do  not 
add  to  the  stability  of  the  body  as  regards  overturning. 


100  THEORETICAL   MECHANICS. 

Stability  of  Equilibrium— Stable ;  Unstable;  Neutral. 

(a)  Pivoted  Bodies.  It  has  been  stated  that  when  a 
body  is  suspended  freely  from  a  pivot  its  center  of  gravity 
assumes  a  position  vertically  below  or  above  the  point  of 
support  ;  otherwise  the  body  is  not  in  equilibrium.  One 
exception  however,  should  be  noted.  If  the  point  of  sup- 
port coincides  with  the  center  of  gravity  the  body  is  at 
equilibrium  in  any  position,  and  will  remain  at  rest 
wherever  placed  by  turning  it  around  the  pivot. 

When  the  center  of  gravity  of  a  pivoted  body  is  verti- 
cally above  the  point  of  support  its  position  is  very  insecure  ; 
any  slight  displacement  of  the  body  will  cause  it  to  roll 
over  with  its  center  of  gravity  downward.  Though  the 
body  was  in  equilibrium  it  lacked  stability,  and  is  said  to  be 
in  a  position  of  unstable  equilibrium. 

But  the  position  it  naturally  assumes,  with  the  center  of 
gravity  vertically  below  the  point  of  support,  is  the  one  of 
maximum  stability.  If  it  is  displaced  from  this  position  it 
returns  to  it  by  force  of  its  own  weight.  It  is  in  stable 
equilibrium. 

When  the  point  of  support  coincides  with  the  center 
of  gravity  the  equilibrium  of  the  body  is  neutral. 

From  these  considerations,  therefore,  we  derive  two  tests 
by  which  to  judge  the  kind  of  equilibrium  of  a  body. 

(i)  Its  equilibrium  is  stable  if  the  body  tends  to  return 
to  the  same  position  after  a  slight  displacement  ;  but  if  a 
slight  jar  or  disturbance  causes  the  body  to  move  still 
farther  from  its  first  position  its  equilibrium  was  unstable  ; 
if  it  remains  wherever  it  is  placed,  its  equilibrium  is  undis- 
turbed by  the  change  and  is  said  to  be  neutral. 

(ii)     When  the  center  of  gravity  is  in  its  lowest  possible 
position  vertically  below  the  point  of  support,  the  body  is  in 


CENTER    OF   GRAVITY. 


IOI 


stable  equilibrium  ;  when  the  center  of  gravity  is  at  its 
highest  possible  position  vertically  above  the  point  of 
support  the  body  is  in  unstable  equilibrium  ;  when  the 
center  of  gravity  coincides  with  the  point  of  support  the 
equilibrium  is  neutral. 

A  third  test  of  the  stability  of  a  body  can  be  deduced 
from  the  consideration  that  any  motion  of  the  body  around 
the  pivot,  which  results  in  elevating  the  center  of  gravity 
requires  the  same  amount  of  work — the  same  expenditure 
of  energy — as  if  the  whole  mass  were  raised  bodily 
through  the  same  distance.  If  a  bar  of  iron,  shown  in 
Fig.  55,  weighs  8  pounds  and  its  center  of  gravity  is  raised 
through  a  vertical  height  of  2  feet  by  rotating  through  the 
arc  CC',  then  the  work  done  is  16 
foot  pounds, — although  a  part  of 
the  rod  has  not  been  raised  at  all. 

(iii)  Now  consider  the  three 
positions  of  the  center  of  gravity 
relatively  to  the  point  of  support — 
vertically  below  ;  vertically  above  ; 
and  coinciding  with  it — and  the 
performance  of  work  involved  in 
any  slight  displacement  from  each 
of  these  positions  of  equilibrium. 
In  the  first  case  the  center  of 
gravity  is  raised  and  hence  work 
is  done  upon  the  body  to  displace  it  from  a  position  of 
stable  equilibrium.  In  the  second  case  the  center  of  gravity 
is  lowered,  and  hence  when  the  body  is  displaced  from  a 
position  of  unstable  equilibrium  work  is  done  by  the  body, 
its  potential  energy  being  converted  into  energy  of  motion. 
In  the  third  case  the  displacement  of  the  body  is  not 
accompanied  by  any  change  in  the  position  of  the  center  of 


2    fi 


55- 


102  THEORETICAL   MECHANICS. 

gravity,  and  hence  a  body  in  neutral  equilibrium  has  the 
same  potential  energy  in  all  positions. 

(V)  Bodies  resting  on  a  base.  A  cone  placed  in  various 
positions  on  a  horizontal  plane  will  afford  typical  illustra- 
tions of  the  three  kinds  of  equilibrium.  Resting  on  its 
base  it  is  stable  ;  balanced  on  its  apex  it  is  unstable  ;  and 
lying  on  its  side — on  an  element— it  is  in  neutral  equilibrium. 
In  general,  if  a  body  rests  (at  equilibrium,  of  course)  on 
a  base  of  any  appreciable  area  it  is  stable,  because  any 
displacement  tending  to  overturn  it,  or  give  it  a  new  base  of 
support,  will  result  in  elevating  the  center  of  gravity,  and 
hence  require  an  expenditure  of  energy.  If  it  rests  on  a 
point  or  line,  as  a  cube  on  a  corner  or  an  edge,  with  its 
center  of  gravity  vertically  above,  in  such  manner  that  any 
displacement  of  overturning  would  lower  the  center  of 
gravity,  it  is  unstable.  But  if  it  rests  on  a  point  or  line  in 
such  manner  that  the  support  may  be  shifted  to  other  points 
or  lines — a  sphere,  or  a  cylinder  on  its  side — without  raising 
or  lowering  the  center  of  gravity,  it  is  in  neutral  equilibrium. 
If  a  body  rests  on  an  area  of  support  with  the  center  of 
gravity  vertically  above  any  point  in  the 
perimeter  of  the  base,  as  in  Fig.  56,  we 
have  a  limiting  case.  Any  slight  dis- 
placement to  the  right  will  prove  the 
present  position  of  the  body  to  be 
one  of  unstable  equilibrium,  while  from 
a  displacement  to  the  left  it  will  recover 
its  present  place  as  if  it  were  stable. 

Examples : 

1.     If  a    rectangular   block    rests    on    one   face  what  kind  of 
equilibrium  does  it  possess  f     Why  ?     How  must  it  be  placed  to  be  in 


CKNTKR   OF   GRAVITY. 


103 


unstable  equilibrium  ? 
equilibrium  f     Why  f 


Could  it  be  placed  in  a  position  of  neutral 


Fig-  57- 


2.  How   must  a  cylinder  be  placed  to  be 
in  stable  equilibrium  f     Could  it  be  placed  in  a 
Position  of  unstable  equilibrium  f 

3.  Can  a  sphere  be  placed  in  a  position  of 
stable  equilibrium  on  a  horizontal  plane  ? 

4.  A  pencil  cannot  be  readily  balanced  on 
its  sharpened  point,    but  if   a   knife- blade   be 
stuck  into  each  side,  it  can  then  be  balanced  on 
its  point   from    an    elevated   support,    as    in 
Fig.  j/.     Explain. 


5.  Fig.  jj~e  is  stable,  while  Fig.  53~d,  having  the  same  base, 
is  not  stable.  Explain. 

Degree  of  Stability.  We  have  observed  that  a  pivoted 
body  has  a  choice  of  but  three  positions  of  equilibrium,  and 
in  each  of  these  positions  we  regarded  it  as  possessing  a 
certain  kind  of  equilibrium.  These  differences  of  kind  we 
expressed  by  means  of  adjectives  which  signified  merely 
the  stability  or  non-stability  of  the  body.  In  algebraic 
notation  we  could  have  said  that  in  one  position  the  body  is 
positively  stable  to  a  certain  degree,  and  in  another  position 
it  is  unstable  or  negatively  stable.  The  numerical  value  of 
this  degree  of  stability  may  be  great  or  small,  depending 
upon  the  weight  of  the  body,  and  the  distance  between  its 
center  of  gravity  and  the  point  of  support.  In  a  position 
of  neutral  equilibrium  this  distance  vanishes  and  the  stability 
is  zero.  It  follows,  therefore,  that  the  equilibrium  of  a 
pivoted  body  may  be  represented  in  all  its  aspects  by  a 
single  algebraic  quantity,  of  which  the  numerical  value  will 
represent  the  degree  of  stability,  and  the  sign  of  quantity 
will  show  the  kind  of  equilibrium. 


104  THEORETICAL   MECHANICS. 

When  a  body  rests  on  a  base  of  support  its  degree  of 
stability  is  measured  by  the  work  necessary  to  overturn  it. 
For  example,  let  us  investigate  the  stability  of  a  rectangular 
block  resting  on  a  horizontal  plane.  Suppose  it  measures 
2  feet  x  3  feet  x  4  feet  and  weighs  150  pounds.  There  are 
six  cases  to  be  considered,  since  the  work  to  be  done  in 
overturning  depends  upon  which  face  the  block  rests  upon, 
and  the  edge  over  which  it  is  turned. 

These  cases  are  as  follows  : 

1.  Resting  on  2  x  3  base. 

(a)  Turned  over  2-foot  edge. 

(b)  Turned  over  3-foot  edge. 

2.  Resting  on  3  x  4  face. 

(#)     Turned  over  3-foot  edge. 
(£)     Turned  over  4-foot  edge. 

3.  Resting  on  2  x  4  face. 

(a)     Turned  over  2 -foot  edge. 

(£)     Turned  over  4-foot  edge. 
Solution : 

i.     Resting  upon  a  2  x  3  face. 
(#)  Turned  on  2-foot  edge. 

If  the  body  starts  from  the  initial  position  A  B  C  D 
(Fig.  58)  and  is  overturned  on  the  two-foot  edge,  its  center 
of  gravity  will  describe  the  arc  O  O'O".  Since  the  edge 
A  B  is  3  feet  and  the  height  B  C  is  4  feet,  the  distance  O  A 

will  be  -  -  or  2 . 5 .    When  the  body  reaches  the  position 

A  B'C'D'  it  occupies  a  position  of  unstable  equilibrium  and 
no  further  work  will  be  required  for  overturning ;  if  it  is 
displaced  to  the  slightest  degree  beyond  this  position  it  will 
then  fall  by  its  own  weight  to  the  position  AB"C"D". 
Therefore,  the  work  necessary  to  overturn  the  body  from  its 


CENTER   OF   GRAVITY. 


105 


first  position  of  stable  equilibrium  is  the  amount  required  to 
raise  the  center  of  gravity  from  O  to  <9', — which,  as  we 
have  already  learned,  is  equivalent  to  raising  the  entire  mass 
through  the  same  distance.  In  describing  the  arc  O  O'  the 


B  3  A  D" 

Fig.  58 

center  of  gravity  is  raised  through  the  vertical  distance 
E  O',  equal  to  0.5  ft.  The  weight  of  the  body  being  150 
pounds,  the  work  done  is  150  X  0.5  or  75  foot-pounds. 

^N  This  gives  us  a  measure  of  the 

x^        stability  of  the   body   as  opposing 
*V    any  effort  to  turn  it  from  the  given 
/      position,    over   the   edge   specified. 
t'         It  would  be  hardly  sufficient,  how- 
f  ever,  to  say  merely  that  the  stability 

'  of  the  body  is  75  foot-pounds  ;  we 

/  must  specify  all  incidental  conditions 

upon  which  this  depends, — the  face 
upon  which  it  rests,  and  the  edge 
Fig.  59.  over  which  it  is  turned. 


io6 


THEORETICAL  MECHANICS. 


(b)  Resting  on  the  same  base  and  turned  on  the  3-foot 
edge  the  vertical  distance  through  which  the  center  of 
gravity  will  be  raised  (Fig.  59)  is  0.236  ft.,  and  the  work 
done  is  35.4  foot-pounds. 

The  remaining  cases  are  to  be  solved  by  the  student  and 
the  results  entered  in  the  following  table  : 


HEIGHT  OF 

WIDTH  OF 

CENTER  OF  GRAVITY 

BASE 

STABILITY 

IN  FEET 

IN  FEET 

a 

2 

3 

75 

b 

2 

2 

35-4 

a 

• 

b 

a 

b 

By  comparing  these  results  it  will  be  seen  that  the  body 
possesses  the  greatest  stability  when  the  center  of  gravity  is 
as  low  as  possible  and  the  base  as  wide  as  possible. 

This  will  be  shown  still  more  clearly  if  the  results  are 
re-arranged  in  proper  places  in  the  table  on  the  following 
page,  in  which  the  width  of  base  zwcreases  in  spaces  from 
left  to  right,  and  height  of  center  of  gravity  decreases  from 
the  top  downward. 


CENTER   OF   GRAVITY. 


I07 


For  any  given  base  the  stability  is  greater  when  the 
center  of  gravity  is  lower;  for  any  given  height  of  the  center 
of  gravity  the  stability  is  greater  when  the  base  is  broader. 


WIDTH  OF  BASF, 

2  FEET 

3  FEET 

4  FEET 

I 

M 

2 

35.4 

75 

0 

FEET 

fib 

O 

w 

H 

W 

1.5 

fc 

FEET 

0 

0 

I 

1 

FOOT 

Examples : 

1.     What  work  is  done  in  overturning  a  cube  of  metal  measuring 
i  foot  each  way,  and  weighing  joo  pounds  ? 


2.  (a)  A  mass  of  metal  measuring  i  foot  x  /  foot  x  2  feet  and 
weighing  1000  pounds  rests  on  one  end.  What  work  is  done  in 
overturning  it? 

(b)  What  work    will  be  done  in  overturning   it  from  a 
position  of  rest  on  one  side,  turning  it  on  the  2-foot  edge  ? 

(c)  What  if  turned  from  the  same  position  on  the  i-foot 
edge? 


io8  THEORETICAL  MECHANICS. 

3.  A  stove  3  feet  high,  2  feet  wide  and  j>  feet  long,  weighing 
joo  pounds,  rests  on  legs  6  inches  high.  Another  weighing  joo 
pounds,  rests  on  legs  12  inches  high  and  measures  j  feet  high,  j  feet 
long  and  j  feet  wide.  Which  stove  is  the  most  stable,  and  in  what 
ratio  f 

NOTE  :  Disregard  weight  of  legs  and  consider  center  of  gravity 
as  being  at  center  of  figure  of  the  "body"  of  stove. 


CHAPTER  IV. 
PRINCIPLES  OF  MACHINES.    THE  LEVER. 

Tools  and  Machines.  By  means  of  a  few  hand  tools  a 
blacksmith  overcomes  the  resistance  of  a  piece  of  iron  and 
forges  it  into  almost  any  desired  shape,  in  a  manner  that 
would  not  be  possible  by  unaided  human  effort.  Similar 
results  are  accomplished  in  any  material,  as  wood  or  stone, 
by  means  of  suitable  tools,  or  with  greater  facility  by  wood- 
working and  stoneworking  machinery  operated  by  power. 
Most  tools  and  machines  of  this  type,  used  for  purposes  of 
construction  and  including  many  of  the  highly  specialized 
devices  used  in  various  factories,  serve  the  purpose  of  cutting 
and  shaping  materials. 

Under  a  different  head  we  could  classify  machines  that 
are  used  for  lifting  loads  or  moving  masses  of  any  kind. 
This  class  would  include  chain  hoists,  cranes,  pumps,  air 
blowers,  elevators,  etc.,  and  in  the  same  list  we  might  even 
include  a  dynamo,  in  which  the  resistance  overcome  is 
something  more  than  the  mere  mass  of  the  moving  parts  and 
the  results  obtained  are  not  so  readily  discerned. 

A  third  class  would  be  machines  that  are  designed  to 
accomplish  delicacy  and  accuracy  of  motion,  without  over- 
coming any  great  resistance,  and  would  include  weaving 
machines,  sewing  machines,  typewriters  and  type-setting 
machines. 

This  is  by  no  means  a  complete  list  or  classification  of 
the  many  different  kinds  of  tools  and  machines  in  use,  but 
it  is  sufficient  for  the  purpose  of  opening  the  way  to  a 
definition  that  will  indicate  the  fundamental  character  or 
essential  nature  of  a  machine. 


110  THEORETICAL   MECHANICS. 

A  machine  is  an  instrument  by  which  a  given  force  is 
made  to  accomplish  a  result  indirectly  that  would  not  be 
possible  by  direct  application  of  the  force  without  the 
intervention  of  such  a  medium.  It  is  a  device  that  serves  to 
modify  a  force  or  motion,  in  magnitude  or  direction,  or  in  both 
respects.  Any  tool,  implement,  device,  contrivance,  instru- 
ment, appliance,  or  apparatus,  by  which  this  is  accomplished, 
directly  or  incidentally,  involves  a  mechanical  principle 
and  is  a  machine,  according  to  our  definition.  Animal 
motions,  even,  are  due  to  the  various  mechanisms  of  which 
the  anatomical  structure  is  composed. 


Efficiency  of  Machines.  As  already  stated,  many 
machines  are  intended  to  lift  loads  and  in  other  ways  over- 
come resistance.  But  even  those  which  are  designed  to 
accomplish  merely  a  delicate  movement  of  any  part  in  a 
certain  direction  cannot  be  operated  except  by  the  applica- 
tion of  energy  supplied  from  some  source  external  to  the 
machine.  If  the  machine  were  without  weight  and  its 
parts  could  move  without  frictional  resistance,  it  would  be 
capable  of  doing  an  amount  of  useful  work  exactly  equal 
to  the  energy  applied.  But  that  is  practically  impossible  ; 
owing  to  the  frictional  resistance,  air  currents,  etc.,  due  to 
the  moving  parts  of  the  machine,  some  of  the  applied 
energy  is  dissipated,  or  frittered  away,  in  the  form  of  useless 
motion,  heat,  sound,  and  at  times  even  light  and  electricity. 
If  a  machine  performs  only  385  foot-pounds  per  second  of 
useful  work  at  the  expense  of  one  horse-power  of  energy 
(550  foot-pounds  per  second)  applied  to  it,  its  efficiency  is 
only  70  per  cent.  The  efficiency  of  a  machine  depends 
upon  its  structural  features  and  the  condition  of  the  bearing 
parts. 


PRINCIPLES    OP    MACHINES.  Ill 

A  complex  machine  may  be  more  efficient  in  some  parts 
than  in  others.  A  machine-shop  planer,  for  instance,  might 
be  conveniently  segregated  into  three  parts,  (i)  the  hori- 
zontal bed ;  (2)  the  driving  pulleys  and  accessory  parts 
conveying  motion  to  the  bed  ;  and  (3)  the  countershafting. 
The  countershafting  may  convey  to  the  machine  proper  96% 
of  the  power  taken  from  the  main  shaft;  the  driving  parts  of 
the  planer  may  furnish  to  the  bed  only  82%  of  the  power 
received  from  the  countershaft ;  and  of  this  the  losses  due 
to  motion  of  the  bed  may  leave  only  88%.  Hence  the 
complete  machine  would  furnish  in  useful  work  only  SS% 
of  82%  of  96%  of  the  power  taken  from  the  main  shaft ;  or 
its  efficiency  would  be  less  than  70%. 

The  Simple  Machines.  Any  mechanical  contrivance, 
howsoever  complicated,  can  be  analyzed  into  certain  elemen- 
tary parts,  commonly  designated  as  the  Mechanical  Powers, 
or  Simple  Machines.  These  are  : 

1.  The   Lever,  (including  the  bent  lever  and   bell 

crank). 

2.  The  Wheel   and   Axle,   (sometimes  classed  as  a 

lever). 

3.  Pulleys. 

4.  Inclined  Plane. 

5.  The  Wedge,  (which  may  be  included  under  the 

inclined  plane). 

6.  The  Screw,  (which  may  also  be  included  under 

the  inclined  plane). 

7.  The  Toggle  Joint,  (sometimes  called  by  analogy 

the  "knee-joint,"  or  ''elbow-joint.") 


112  THEORETICAL   MECHANICS. 

All  the  elementary  or  component  parts  of  any  machine 
could  be  classified  under  these  heads.  The  most  complicated 
machine  is  nothing  more  than  an  assemblage  of  parts 
involving  combinations  and  modifications  of  these  elemen- 
tary mechanisms. 


The  Lever.  A  lever  is  a  rigid  rod  free  to  move  about  a 
single  fixed  point  called  the  fulcrum.  It  is  the  simplest  form 
of  a  machine,  and  from  it  we  can  deduce  the  fundamental 
principles  and  considerations  of  all  the  mechanical  powers. 


(/)  Principle  of  Virtual  Work,  or  Virtual  Velocities. 
If  a  weight  W  (Fig.  60),  suspended  from  one  end  of  a 
rigid  rod,  is  balanced  by  a  force  P  acting  vertically  down- 
ward on  the  other  end,  or  on  any  point  beyond  the  fulcrum, 
the  relative  values  of  P  and  W  necessary  for  equilibrium 
depend  upon  the  distances  a  and  b,  or  AF  and  BF. 


A 


Fig.  60. 

The  two  forces  are  inversely  proportional  to  the  distances 
from  their  points  of  application  to  the  fulcrum. 


For  example,  if  the  short  arm  of  the  lever  is  3  feet  long  and 
the  other  arm  5  feet,  a  weight  of  i  o  pounds  suspended  from 


PRINCIPLES    OF    MACHINES.  113 

the  end  of  the  short  arm  would  be  balanced  by  a  force  of 
10  X  -!-,  or  6  pounds,  applied  vertically  downward  at  the 
other  end. 

This  can  be  proved  by  the  principle  of  Virtual  Work. 
Assuming  that  the  rod  is  without  weight  and  that  P  and  W 
produce  equilibrium,  suppose  that  the  rod  be  given  any 
displacement  through  an  angle  y  (Fig.  61),  in  either 
direction  around  the  fulcrum.  If  the  weight  is  raised  to  a 
position  W\  through  a  vertical  height  h,  the  other  end  falls 
through  a  vertical  distance  k.  Now  h  —  a  sin  y,  and 

k  -  b  sin  y  ;  whence  — -  =  -7-.     If  h  =  i  foot,  k  is  therefore 
k          b 

equal  to  I  feet.      Any  displacement  of  the  rod  which  raises 


k\\ 

-J/y 


Fig.  61. 


the  weight  one  foot  requires  an  expenditure  of  10  foot- 
pounds of  energy.  But,  as  we  have  shown,  this  displacement 
of  the  weight  would  be  accompanied  by  a  falling  of  the 
other  end,  or  a  movement  of  the  applied  force,  through  a 
distance  i  feet,  and  since  the  applied  force  is  6  pounds, 
the  energy  furnished  by  it  is  6  X  -3-,  or  10  foot  pounds. 
The  work  done  by  the  applied  force  is  therefore  just  equal 
to  the  work  done  upon  the  weight,  and  on  the  whole  there 
is  no  gain  or  loss  of  energy.  This  is  strictly  in  accord  with 
the  Law  of  the  Conservation  of  Energy,  and  is  applicable 
as  a  test  of  the  equilibrium  of  any  of  the  mechanical  powers. 
That  is,  to  determine  whether  the  applied  force  is  just 
sufficient  to  balance  the  weight  or  resistance,  we  imagine  a 


114  THEORETICAL   MECHANICS. 

slight  displacement  of  the  machine  in  such  manner  as  it  is 
free  to  move,  and  then  calculate  the  product  of  the  weight 
times  the  vertical  distance  through  which  it  moves.  This 
should  be  equal  to  the  product  of  the  applied  force  times  its 
vertical  distance ;  or,  referring  to  the  figure,  we  should  have 

Wh=Pk.  (10) 

This  is  called  the  Principle  of  Virtual  Work,  or  Virtual 
Velocities.  The  adjective  "  virtual  "  is  used  to  signify  that 
there  is  no  real  motion  or  displacement,  and  no  work  is 
actually  performed  either  upon  the  weight  or  by  the  applied 
force  ;  but  the  result  attained  by  the  supposition  that  motion 
does  take  place,  and  that  work  is  done,  is  virtually  the  same 
in  effect  as  if  the  machine  really  moved,  in  spite  of  the  fact 
that  it  is  in  equilibrium. 

W       k  k          h 

If  Wh  =  Pk,  then  -£  =  -£-•     Since  -y-  =  — ,    we  have 
P         h  ha 

W       b 
—p  -  — ,  thus  proving  our  proposition,  that  two  balancing 

forces  applied  to  a  lever  are  inversely  proportional  to  the 
distances  of  their  points  of  application  from  the  fulcrum. 


(//)  Principle  of  Moments.  In  the  case  under  considera- 
tion, where  the  two  forces  act  at  right  angles  to  the 
lever  on  opposite  sides  of  the  fulcrum,  the  distances  a  and  b 
are  called  the  lever  arms.  The  product  of  a  force  times  its 
leverage,  or  lever  arm,  is  called  the  moment  of  the  force. 
A  force  acting  at  the  end  of  an  arm  2  feet  long  has  twice 
the  advantage  of  the  same  force  if  its  lever  arm  is  only  one 
foot. 

The  Principle  of  Moments  asserts  that  two  forces  acting 
on  a  lever  are  in  equilibrium  if  their  moments  are  equal 


PRINCIPLES    OF    MACHINES.  115 

and  tend  to  turn  the  lever  in  opposite  directions.  The 
demonstration  of  this  principle  requires  only  a  simple 

W        b 
transformation    of    the     expression   —  :        -  to   the  form 

W  a  —  P  b.  In  the  case  under  consideration  (Fig.  60)  the 
weight  10  pounds,  with  a  leverage  of  3  feet  is  exactly 
balanced  by  the  opposing  force  of  6  pounds  with  a  leverage 
of  5  feet,  because  both  have  moments  of  30  units. 

A  moment  is  usually  designated  as  so  many  pound-feet, 
or  ton-feet,  for  obvious  reasons.  These  should  not  be  con- 
founded with  the  unit  of  work,  the  foot-pound.  In  both  cases 
we  use  .a  compound  word  made  up  of  a  unit  of  force  and  a  unit 
of  distance,  but  the  order  of  combining  the  component  words 
is  different  in  the  two  cases,  and  furthermore  the  unit  of 
distance  represents  a  leverage  in  one  case  while  in  the  other 
case  it  refers  to  the  distance  through  which  a  weight  is 
raised  or  other  resistance  overcome. 


\AAFB\B                L 

E 

p 

00                      V                     *" 
P 

\ 

Fig.  62. 


If  several  forces  and  weights  act  on  the  same  lever 
simultaneously,  the  sum  of  the  moments  of  the  applied  forces 
must  be  equal  to  the  sum  of  the  moments  of  the  weights, 
to  produce  equilibrium.  And  if  any  of  the  forces  act  in  a 
direction  tending  to  aid  the  weights,  such  forces  should  be 


Il6  THEORETICAL   MECHANICS. 

given  a  negative  moment.     In  Fig.  62,  the  forces  and  weights 
are  in  equilibrium  provided 

IV  X  AF  +  W'X  A'  F  +  W"  XA"F  = 
PX  B  F-P'X  B'  F+P"X  B"  F+  P'"  X  B'"  F. 

Moment  Due  to  Weight  of  Lever.  A  rod  of  uniform 
dimensions  supported  midway  between  the  two  ends  is  in 
equilibrium,  because  its  center  of  gravity  is  in  vertical  line 
with  the  point  of  support,  (see  p.  98).  But  if  the  fulcrum 
is  not  at  the  center,  as  illustrated  in  Fig.  63,  the  weight  of 
the  rod  acts  like  a  load  applied  at  its  center  of  gravity  ;  as 
if  the  entire  weight  W  of  the  rod  were  concentrated  at  that 
point.  Hence,  if  such  a  rod  were  used  for  a  lever,  we 


TV 


Fig.  63. 

could  allow  for  its  weight  by  considering  its  effect  equal  to 
moment  W  X  C  F.  If  the  center  of  gravity  is  over  the 
fulcrum,  this  moment  is  zero  and  the  weight  of  the  rod  may 
be  disregarded. 

Sometimes  it  is  easy  to  allow  for  the  weight  of  the  rod  by 
proportioning  the  weights  of  the  parts  on  opposite  sides  of  the 
fulcrum,  and  multiplying  the  weight  of  each  part  by  the  distance  of 
its  center  from  the  fulcrum,  as  illustrated  in  Fig.  64,  in  which  W^  and 
W2  are  the  weights  of  the  portions  of  the  lever  on  opposite  sides  of 
the  fulcrum.  If  the  weight  of  each  part  is  assumed  to  be  massed  at 
its  center,  then  the  heavy  rod  is  equivalent  to  a  weightless  rod  with 
the  weights  1Vl  and  Wz  suspended  from  C\  and  C2  respectively. 


PRINCIPLES   OF   MACHINES.  1 17 

This  method  seems  to  suggest  itself  to  most  persons  more  readily 
than  the  more  direct  method  previously  described,  but  it  is  a  round- 
about procedure  and  sometimes  involves  difficulties. 


F  | 

r*tf  r  j£i 

Fig.  64. 
Examples: 

1.  A  heavy  rod  balanced  on  a  fulcrum  at  its  middle  point  has  a 
weight  of  40  pounds  suspended  from  a  point  2  feet  from  the  fulcrum. 

(a)  At  what  distance  on  the  other  side  of  the  fulcrum  must 
an  iS-pound  weight  be  placed  in  order  to  produce  equilibrium  ? 

(b]  What  weight  would  have    been   sufficient   to  produce 
equilibrium  if  it  had  been  placed 3.5  feet  from  the  fulcrum  f 

2.  A   heavy  rod  j  feet  long  is  balanced  on   a  fulcrum  at  its 
middle  point.      If  a  weight  of  10  pounds  is  suspended  from  one  end 
and  a  second  weight  of  5  pounds  is  placed  at  a  point  one  foot  from 
the  fulcrum  on  the  same  side,  what  downward  pull  must  be  exerted 
on  the  other  end  of  the  rod  to  produce  equilibrium  ?       What  if  the  j- 
pound  weight  had  been  placed  one  foot  from  the  fulcrum  on  the  side 
towards  the  downward  pull? 

3.  If  a  stick  of  timber  9  feet  long  and  weighing  16  pounds  is 
supported  on  a  fulcrum  j  feet  from  one  end,  what  weight  must  be 
suspended  from  this  end  to  produce  equilibrium  f     Where  would  a 
12-pound  weight  be  placed  to  produce  the  same  result  ? 

4.  A  uniform  iron  rod  n  feet  long  and  weighing  40  pounds  is 
supported  on  a  fulcrum  j.j  feet  from  one  end.     If  a  1 4-pound  weight 
is  hung  from  this  end,  where  must  a  second  weight  of  28  pounds  be 
placed  to  produce  equilibrium  ? 

5.  For  the  Principle  of  Work  we  found  that  Wh  =  Pk,  and  for 
the  Principle  of  Moments  Wa  =  Pb.      Explain  the  difference  between 
these  principles  by  contrasting  the  meanings  of  h  and  k  with  a  and  b. 


Il8  THEORETICAL   MECHANICS. 

Three  Kinds  of  Levers.  The  applied  force  and  the  load 
acting  on  a  lever  are  not  always  on  opposite  sides  of  the 
fulcrum.  If  a  rod  is  fixed  at  one  end,  as  shown  in  Figs.  65 


A 


and  66,  a  weight  acting  at  any  point  may  be  balanced  by  a 
force  exerted  at  any  other  point.  The  relative  values  of 
P*  and  W*  depend  upon  the  distances  AF  and  BF,  exactly 


V 


Fig.  66. 

as  determined  for  the  lever  of  the  first  kind,  from  which  we 
deduced  the  Principle  of  Work  and  the  Principle  of  Moments. 
Whatever  the  relative  positions  of  P,  W  and  F,  these 
principles  apply  with  equal  strictness.  In  Fig.  65,  since 
W  X  AF  =  P  X  BF,  and  BF  is  greater  than  AF,  the 
applied  force  is  less  than  the  weight.  In  Fig.  66  it  is 
greater. 


*The  employment  of  the  letters  P  and  W,  to  represent  respectively  the 
Applied  Force  and  the  Weight,  has  become  a  matter  of  conventional  usage.  It  is 
assumed,  of  course,  that  W  does  not  always  stand  for  a  weight  or  gravitational 
action,  but  may  be  a  load,  or  a  tension,  or  resistance  of  any  kind  whatsoever.  The 
applied  force  Pis  frequently  called  the  "Power;"  whence  the  symbol  P.  This  use 
of  the  word  "Power,"  referring  to  a  force  merely,  is  not  consistent  with  our 
previous  acceptance  of  its  meaning  in  the  sense  of  "Rate  at  which  a  machine  can 
do  work."  For  the  sake  of  accuracy  we  shall  adhere  to  the  expression  "Applied 
Force,"  symbolized  by  the  capital  P. 


PRINCIPLES   OF   MACHINES.  1 19 

If  P  and  W  are  on  opposite  sides  of  the  fulcrum,  the 
lever  is  said  to  be  of  the  "first  order."  Fig.  65  illustrates  a 
lever  of  the  "second  order,"  and  Fig.  66  one  of  the  "third 
order." 

In  levers  of  the  first  order  the  applied  force  may  be 
greater  or  less  than  the  load.  In  levers  of  the  second  order 
the  applied  force  is  always  less  than  the  load,  while  in 
levers  of  the  third  order  the  load  is  less. 

Compound  Levers.  In  some  devices  —  certain  gate 
latches  and  wagon  brakes,  for  examples, — a  train  of  levers 
is  used  to  multiply  the  force  or  motion,  as  the  case  may  be. 


*. 


,  B 

Fig.  67. 

As  shown  in  Fig.  67  the  weight,  or  resistance,  of  one  is 
propagated  to  the  next  as  an  applied  force.  It  is  not 
necessary  that  all  in  the  series  be  levers  of  the  same  order. 

Examples  of  compound  levers  are  found  in  platform 
scales,  typewriting  machines,  piano  keys,  the  trigger  of  a 
gun,  and  in  railroad  switches. 

Examples : 

1.  To  which  class  of  levers  would  you  assign  each  of  the  follow- 
ing devices: 

Pair  of  pliers ;  sugar  tongs;  scissors ;  nut  cracker ;  black- 
smith's vise  ;  blacksmith's  tongs. 

2.  Name  five  devices  in  which  the  lever  is  used.     At  least  one  of 
the  five  must  refer  to  the  lever  of  the  third  order. 

3.  How  many  of  the  three  kinds  of  levers  are  illustrated  in  the 
ordinary  use  of  a  crowbar  ? 


120  THEORETICAL   MECHANICS. 


Safety  Valves.  The  ordinary  safety  valve,  used  as  a 
boiler  attachment  to  prevent  explosions,  is  a  lever  of  the 
third  order.  A  valve  V  (Fig.  68)  in  direct  communication 
with  the  interior  of  the  boiler,  is  pivoted  at  B  to  a  bar  AF. 


Fig.  68. 

Whenever  the  upward  pressure  on  V  reaches  a  certain  limit 
sufficient  to  overcome  the  resistance  due  to  the  weight  of 
W  and  of  other  parts  of  the  mechanism,  the  lever  and  valve 
are  moved  upward,  turning  on  F  as  a  fulcrum.  By  this 
means  a  part  of  the  contents  of  the  boiler  is  allowed  to 
escape,  and  if  by  that  means  the  boiler  pressure  is 
sufficiently  reduced,  the  valve  drops  back  into  its  seat. 

Unless  the  area  of  the  valve  is  sufficient  to  allow  the 
steam  to  escape  as  fast  as  it  is  generated,  perfect  safety  is  not 
secured.  As  the  generating  capacity  of  the  boiler  depends 
mainly  upon  the  area  of  the  grate  surface,  it  is  customary  to 
allow  one  square  inch  of  valve  area  for  every  two  square 
feet  of  grate  surface. 

Examples : 

1.  How  would  you  find  the  total  upward  pressure  on  the  valve? 
Should  you  allow  for  the  atmospheric  pressure  on  the  top  of  the 
valve  ? 

NOTE:  The  reading  of  a  steam  gauge  indicates,  not  the 
actual  boiler  pressure,  but  the  difference  between  this  and  the 
pressure  of  the  atmosphere. 


PRINCIPLES   OF   MACHINES.  121 

2.     What  pressure  per  square  inch  (by  gauge']  would  be  necessary 
to  raise  a  safety  valve  constructed  and  adjusted  as  follows: 

Weight  of  valve,  8  pounds. 

Diameter  of  valve,  j>  inches. 

Valve  pivoted  4  inches  from  fulcrum. 

Weight  of  lever  12  pounds  (uniform  rod}. 

Total  length  of  lever,  2  feet. 

Weight  of  150  pounds  suspended  from  e.rtreme  end  of 

lever. 
All  as  illustrated  by  diagram,  Fig.  69. 


B 


Fig.  69. 

3.  At  what  point  on  this  lever  arm  would  it  be  necessary  to  place 
a  2oo-pound  weight  to  withstand  the  same  pressure  ? 

4.  At  what  point  must  the  ijo  pound  weight   be  placed  to 
balance  a  boiler  pressure  of  100  pounds  per  square  inch  (by  gauge}? 

5.  What  weight  must  be  placed  at  the  end  of  the  rod  to  balance  a 
boiler  pressure  of  go  pounds  per  square  inch  (by  gauge]? 

6.  If  a  boiler  is  over  a  grate  5  feet  square,  what  diamzter  of 
safety  valve  should  it  have  ? 


Pressure  on  the  Fulcrum— Parallel  Forces,  (a)  fevers 
of  the  First  Order.  If  a  ball  weighing  10  pounds 
(Fig.  70)  is  fastened  to  one  end  of  a  weightless 
lever,  3  feet  from  the  fulcrum,  it  can  be  balanced 
by  a  second  ball  weighing  6  pounds  and  fastened 
on  the  other  end  at  a  distance  of  5  feet  from  the 
fulcrum.  It  is  obvious,  in  this  instance,  that  the  fulcrum 


122 


THEORETICAL   MECHANICS. 


bears  the  entire  load  of  16  pounds,  W-\-P.  And  since 
the  entire  system  of  combined  masses  is  at  equilibrium  its 
center  of  gravity  must  be  in  vertical  line  with  the  point  of 
support,  as  if  the  entire  weight  (16  pounds)  were  acting 
downward  on  that  point. 

Since  the  distances  of  the  fulcrum  from  A  and  B  are 
inversely  as  the  weights  acting  at  those  points,  it  follows  as 
a  general  principle  that  if  two  parallel  forces  in  the  same 


A  v%0 

Fig.  70. 

direction  act  on  two  different  points  of  a  body,  their  resultant 
is  equivalent  to  a  single  force  equal  to  their  sum  and  acting 
at  a  third  point  whose  distance  from  the  two  points  of 
application  is  inversely  as  the  forces. 

If  more  than  two  parallel  forces  are  acting  simultaneously 
on  the  same  body,  the  resultant  of  any  two  may  first  be 
found,  and  this  may  then  be  combined  with  a  third  force,  etc. 

Examples : 

1.  A  rectangular  block  of  wood  I  cm.  x  /  cm.  x  <?.j  cm.  is  glued 
to  a  cube  of  lead  i  cm.  x  /  cm.  x  /  cm.,  in  the  manner  shown  in 


\          7 

v 

~~~---.         ,--'"' 

'K~ 

:;;:A:-"""---._ 

\ 

'  \  VK 

1 
Wood               \Lead 

I 

6 

r--i 

Fig   71.  Fig.  72. 

71.      Locate  the  center  of  gravity  of  the  combined  masses. 
(Specific  gravity  of  lead  -  11.3;  specific  gravity  of  wood  -  0.6}. 

2.  A  %-inch  bolt  has  a  head  |  inch  square  and  ^  inch  high.  If 
the  shank  of  the  bolt  is  2  inches  long,  locate  the  center  of  gravity  of 
the  entire  bolt. 


PRINCIPLES   OF    MACHINES.  123 

(£)  Levers  of  the  Second  Order.  In  a  lever  of  the 
second  order  the  pressure  on  the  fulcrum  is  W—P.  For 
example,  if  a  weight  of  12  pounds  is  suspended  at  a 
distance  of  i  foot  from  the  fulcrum,  it  can  be  balanced  by  a 


F 

A 

I 

A 

1^ 

B 

On 

Fig-  73- 

force  of  4  pounds  if  the  latter  is  applied  at  a  distance  of 
BF  -  3  feet.  In  this  case  the  fulcrum  supports  a  load  of 
8  pounds. 

To  make  this  clear,  imagine  that  the  stick  is  supported 
by  two  persons,  each  having  an  end  of  the  stick  resting  on 
one  of  his  shoulders,  and  the  1 2-pound  weight  being 
suspended  as  stated,  one  foot  from  one  end  and  two  feet  from 
the  other.  The  person  holding  the  end  B  will  support 
4  pounds,  leaving  the  person  at  F  to  hold  8  pounds. 
Because,  if  the  person  at  B  should  raise  his  shoulder  the 
lever  would  turn  about  F  as  a  fulcrum,  and  as  the  point  B 
would  describe  an  arc  with  radius  FB>  while  A  moves  with 

ZfjD 

radius  FA  =       — ,  then  by  the  principle  of  work  the  end  B 

\) 

W 

would  be  lifted  upward  by  a  force  P  -     — ,  or  4  pounds.     If, 

o 

on  the  contrary,  the  person  at  F  should  raise  his  end  of  the 
rod,  moving  it  about  B  as  a  fulcrum,  the  arcs  described  by 
F  and  A  would  have  radii  of  3  feet  and  2  feet, 
respectively,  whence  the  upward  force  exerted  at  F  would 

2 

be  --X  W,  or  8  pounds. 


124  THEORETICAL   MECHANICS. 

If  we  choose  to  apply  the  principles  of  parallel  forces, 
we  can  say  that  whatever  the  resistance  which  the  fulcrum 
is  required  to  exert  it  is  equivalent  to  a  force  P'  (Fig.  74) 
acting  vertically  upward.  Then  if  the  rod  is  in  equilibrium 
the  resultant  of  the  two  parallel  forces  P  and  P'  must  be 
equivalent  to  the  single  force  of  1 2  pounds  acting  at  point 


A 


A 


Fig.  74- 

A, — a  force  equal  to  W  but  opposite  in  direction.  And  the 
moments  of  P  and  P'  about  A  must  be  equal,  or  Pf  X  i  = 
4X2,  whence  P' =  8. 


(c)     Levers  of  the  Third  Order. 

Exercise : 

Prove  that  in  levers  of  the  third  order  the  pressure  on  the  fulcrum 
is  exerted  in  the  direction  of  the  applied  force,  and  is  equal  to  P — W. 


Parallel  Forces.  The  Couple.  As  a  rule,  but  not  always, 
two  parallel  forces  acting  on  a  body  at  different  points  can  be  counter- 
balanced by  a  third  force.  If  the  two  parallel  forces  have  the  same 
direction,  as  P  and  Pl  in  (Fig.  75),  the  resultant  is  a  force  R  equal 
to  their  sum  and  acting  at  a  point  F  such  that  B  F  :  A  F  :  :  P  l  :  P^ 


PRINCIPLES   OF   MACHINES. 


125 


This  case  is  illustrated  in  levers  of  the  first  order,  as  was  fully  explained 
on  pp.  121  and  122.  If  P  and  P1  are  equal,  F  is  midway  between 
A  and  /?,  as  exemplified  in  a  beam  balance  having  equal  arms. 


Fig.  75- 

If  the  two  forces  are  parallel  and  in  opposite  directions,  as  in 
Fig.  76,  the  resultant  is  a  force  equal  to  the  difference  between  the 
two  and  acting  at  a  point  F,  in  the  line  A  B  continued,  such  that 


Fig.  76. 


A  F  :BF: :  P\Pl.  If  /*,  had  been  greater  than  P  the  point  F 
would  have  been  on  the  side  next  to  A, — in  B  A  continued  instead  of 
A  B.  This  applies  to  levers  of  the  second  and  third  orders,  pp.  123 
and  124. 

If  P  and  Pl  are  equal  and  opposite  they  have  no  real  resultant. 
Two  such  forces  acting  on  a  body  at  different  points  constitute  what 
is  called  a  Couple.  No  single  force  howsoever  applied  could  exactly 
counterbalance  them,  and  hence  they  have  no  resultant.  This  will 
become  evident  if  we  observe  the  changes  that  take  place  in  Fig.  76, 
as  we  substitute  different  values  for  P  and  P1}  finally  making  them 
become  equal  to  each  other.  From  the  relation  A  F  :  B  F  :  :  P  :  P^ 
it  will  be  seen  that,  if  Pl  in  Fig.  76  should  become  smaller  or  P 
become  larger,  the  point  F  will  in  either  event  approach  nearer  to  B, 


126 


THEORETICAL   MECHANICS. 


And  conversely,  if  P  becomes  less,  relatively  to  Pl  — that  is,  if  P  and 
Pl  become  more  nearly  equal  to  each  other — the  point  F  will  recede 
from  B.  A  few  simple  computations  will  show  that  when  P1  finally 
becomes  equal  to  P  the  point  F  recedes  to  an  infinite  distance. 
For  that  purpose  let  us  substitute  a  few  assumed  values  in  the 
expression 

AF         P 


If  P  =   2  Plt 

If  P=  3^i, 
If  P  =  SP1, 

If  P  =  10  P, 


BF  - 

A  F  =  2  B 
A  F  = 
AF  =  „ 
y2F  -  lO.ff.F, 


or  BF  =  AB* 

or  £.F  =  0.5,4  5 

or  B  F  =  0.25  A  B 

or  B  F  =0.11  A  B 


If  P  =  looooo  P!,   A  F  =  100000  B  F,    or    B  F  =  o  ooeoi  A  B 


IfP=  1.5 

If   P   =    I.I 

If  P  =   I.OI 
If    P  =   I.COOOI    P 


AF  =  i.5£F,  or  B  F  =  -2  A  B 

A  F  =  i  i  B  F,  or  B  F  =  10  A  B 

A  F  =  i. 01  B  F,  or  B  F  =  looAB 

A  F  =  I.OOODI  Z?  /•',  or  B  F  =  100000  A  B 


If  the  fractional  difference  between  P  and  Pl  becomes  infini- 
tesimally  small,  then  B  F  becomes  infinitely  greater  than  A  B. 

In  some  branches  of  applied  mechanics  the  idea  of  the  Couple  is 
frequently  met  with,  although  we  shall  not  need  to  use  it  in  develop- 
ing our  subject  from  an  elementary  standpoint.  It  is  illustrated  in 
turning  the  handles  of  a  copying-press,  if  equal  forces  are  exerted  on 
the  two  ends ;  likewise  in  cutting  a  thread  on  a  pipe  or  bolt. 

On  account  of  its  peculiar  mathematical  conditions  the  Couple 
possesses  some  very  characteristic  and  striking  properties.  The  sum 
of  the  moments  of  the  two  forces  involved  in  a  given  couple  is  a 
constant  quantity,  being  the  same  with  reference  to  any  point  that 
may  be  selected,  whether  within  or  without  the  body  acted  upon  by 
the  couple,  and  if  the  body  be  pivoted  so  as  to  move  around  this 
point  under  the  influence  of  the  couple,  the  latter  will  cause  no 
pressure  whatsoever  on  the  pivot  or  fulcrum. 


*  Combining  AF=1BF  with  AF=Ati+BF(?fg.  76),  weget'2BF=AB     BF, 
or  B  F=  A  B.    Solve  the  other  cases  in  the  same  manner. 


PRINCIPLES    OF    MACHINES. 


127 


Bent  Levers.    If  A  F  B  (Fig.  77  or  78)  is  a  rigid  body, 
free  to  turn  about  point  F,   the  forces  P  and    W  will  be  in 


Fig.  77. 


Fig.  78. 


equilibrium  if  W  X  AF  -  P  X  BF.  No  matter  what  the 
shape  of  the  lever,  nor  where  the  fulcrum  is  situated,  if  PB 
and  WA  are  perpendicular  to  BF  and  AF  respectively,  the 
principle  of  moments  applies  exactly  as  if  AF  and  BF 
were  in  the  same  straight  line. 

Moment  of  a  Force  Acting  Obliquely  on  a  Lever.    Referring 
to  Fig.  79,  the  moment  of  P  with  reference  to  the  fulcrum 


A 


Fig.  79. 

F  is  not  P  <  BF.  By  resolving  P  into  two  components, 
pl  and  pn  (Fig.  80),  perpendicular  and  parallel  to  BF,  it 
becomes  apparent  that  the  component  pQ  can- 
not have  any  effect  in  turning  the  lever  ;  it 
simply  pushes  the  lever  in  the  direction  BF 
against  the  bearings  at  F,  without  producing 
rotation.  The  component />n  at  right  angles 


128  THEORETICAL   MECHANICS. 

.to  FB,  is  alone  effective  in  balancing  W.  Whence,  if  the 
lever  is  at  equilibrium  W  X  AF  =  />,  X  BF. 

If  the  angle  PBF  is  called  ft,  then  the  component 
pl  =  P  sin  (3,  whence  W  X  AF  =  P  sin  ft  X  BF. 

Instead  of  resolving  the  force  P  into  two  components,  we 
might  have  turned  to  the  idea  of  the  bent  lever  by  project- 
ing the  arm  FB  into  a  direction  at  right  angles  to  P.  The 
force  P  acting  at  an  angle  ft  with  a  lever  arm  BF  has  the 
same  effective  moment  as  if  it  were  acting  at  right  angles  to 
the  arm  B,  F,  in  Fig.  81.  For,  since  FB,  =  FB  sin  ft,  it 


B 


Fig.   81. 

must  follow  that  W^AF=PXFB  sin  ft.  Therefore, 
when  any  force  acts  obliquely  upon  a  lever,  the  effective 
moment  of  the  force  may  be  found  in  either  of  two  ways  : 
(i)  By  resolving  the  force  into  two  components  perpendicular 
and  parallel  to  the  lever  a*m  ;  or  (2)  by  projecting  the  lever 
arm  into  a  direction  perpendicular  to  the  force.  In  either 
case  the  effective  moment  of  the  force  is  P  X  BF  X  sin  ft. 

It  should  be  noted  that  the  forces  exerted  to  produce 
equilibrium  in  a  bent  lever,  and  likewise  oblique  forces  on  a 
straight  lever,  will  give  rise  to  a  pressure  on  the  fulcrum 
that  is  not  merely  the  sum  or  difference  of  P  and  W.  In 
Fig.  79  for  example,  a  part  of  P  (component pQt  Fig.  80) 
pushes  the  lever  in  the  direction  B  F.  The  pressure  on  F, 
in  this  instance  would  be  the  resultant  of  Pl  -f-  W  acting 
vertically  and  pt>  acting  horizontally. 


PRINCIPLES    OF    MACHINES.  I2Q 

From  these  considerations  of  oblique  forces  it  follows 
that  the  Principle  of  Moments,  deduced  on  pp.  114  and  115 
for  parallel  forces,  has  a  much  more  general  application 
than  was  there  assumed.  If  any  number  of  forces  in  the 
same  plane  act  upon  a  body  to  produce  equilibrium,  the 
sum  of  the  moments  of  these  forces  about  any  point  in  that 
plane  is  equal  to  zero*  The  point  with  reference  to  which 
the  moment  is  taken  need  not  be  within  the  body,  and  the 
forces  need  not  be  parallel,  nor  is  it  necessary  in  our 
diagrams  to  represent  the  form  or  outline  of  the  body  ;  the 
single  line  that  we  have  used  to  connect  the  points  of 
application  of  the  forces  is  sufficient  for  all  necessary 
computations. 

In  the  lever  shown  in  Fig.  82  the  fulcrum  is  called  upon  to 
offer  a  resistance  of  16  pounds,  which  is  equivalent  to  a  force 

A 

i 


Fig.  82. 

of  1 6  pounds  acting  upward,  as  if  a  siring  tied  at  F  were 
pulled  upward  by  such  a  force.  Now,  if  we  take  any  point 
as  A  the  moments  of  the  three  forces,  6,  16  and  10  with 
reference  to  this  point  will  be* 

6  X  8  —  16  X  3  HI  10  X  o  =  o. 
With  reference  to  F  the  sum  of  the  moments  will  be 

6  X  5  —  10  X  3  +_  16  X  0  =  0. 

With  reference  to  a  point  X  two  feet  from  B  the  sum  of  the 
moments  will  be 

6  X  2  -f  16  X  3  —  10  X  6  =  o. 

*A  force  tending  to  produce  motion  in  the  direction  of  the  hands  of  a  clock 
around  the  point  with  reference  to  which  the  moment  is  being  found  is  considered 
.as  having  a  positive  moment;  counter-clockwise  a  negative  moment. 


130  THEORETICAL   MECHANICS. 

Examples : 

1.  Referring  to  Fig.  82,  select  any  other  point  in  the  rod  and 
show  that  with  reference  to  this  point  the  sum  of  the  moments  of  the 
forces  involved  is  zero. 

2.  Prove  the  same  for  a  point  at  any  given  distance  beyond  either 
end  of  the  rod. 

3.  Take  a  case  of  a  couple  of  any  given  numerical  value,  and 
find  the  sum  of  the  moments  of  the  tin' o  forces  with  reference  to  several 
different  points, — say,  one  point  in  the  line  connecting  the  two  forces; 
one  in  this  line  continued;  and  a  third  one  at  some  convenient  place 
entirely  without  this  line.     Is  the  sum  of  the  moments  the  same  in  the 
three  cases  ? 

If  the  two  forces  acton  a  body  to  produce  equilibrium  the  sum  of 
the  moments  of  the  forces  is  zero  for  any  point  and  for  all  points. 
Can  a  couple  produce  equilibrium  / 


•Mechanical  Advantage.  If  a  lever  is  used  to  enable  a 
force  of  6  pounds  to  balance  a  resistance  of  10  pounds,  there 
is  an  obvious  advantage,  but  there  is  also  the  disadvantage 
that  if  motion  takes  place  the  load  will  not  be  moved  as  far 
as  the  applied  force.  Conversely,  if  a  machine  is  contrived 
so  as  to  multiply  the  motion,  as  would  be  the  case  if  a  force 
were  applied  to  the  short  end  of  a  lever  of  the  first  order 
to  move  a  load  at  the  longer  end,  and  always  in  levers  of 
the  third  order,  then  the  motion  gained  would  be  at  the 
expense  of  force  exerted,  for  the  applied  force  would  then 
have  to  be  larger  than  the  load  in  proportion  as  its  leverage 
is  less.  We  can  choose"  either  advantage  but  we  cannot 
accomplish  both  in  the  same  machine  ;  as  already  shown 
(pp.  114)  Wh-  Pk,  so  that  the  applied  force  and  the  load 


PRINCIPLES    OF    MACHINES-  131 

cannot  be   otherwise  than    inversely    proportional    to    the 
distances  through  which  they  would  move. 

Some  mechanical  devices  merely  change  the  direction  of 
the  force  or  motion,  without  gain  or  loss  of  either.  Take, 
for  examples,  the  bell-crank  and  the  single  pulley,  or  even 
the  lever  with  equal  arms. 


Fig.  83. 


The  bell  crank  shown  in  Fig.  77,  changes  the  direction 
of  the  force  ninety  degrees,  but  if  the  two  arms  are  equal, 
the  force  exerted  is  unchanged  in  magnitude. 

The  single  fixed  pulley  (Fig.  83)  permits  no  possible 
gain  or  loss  of  force,  except  by  friction,  but  it  can  be  used 
to  accomplish  any  change  of  direction. 


CHAPTER  V. 


MACHINES. 


THE  WHEEL-AND-AXLE.  A  pilot-wheel  will  serve  as  a 
convenient  type  of  the  numerous  devices  in  which  the  idea 
the  wheel-and-axle  is  employed.  A  comparatively  small 
force  applied  to  the  handles  on  the  periphery  of  the  wheel 
is  sufficient  to  overcome  a  greater  resistance  on  the  part  of 
the  tiller-ropes. 

A  load  W  applied  at  the  axle  or  drum  represented  by 
the  small  circle  in  Fig.  84  has  a  moment  of  W  X  r  with 


Fig.  84. 


Fig.  85. 


reference  to  the  center  F.  If  the  "wheel"  has  a  radius  r ,, 
the  applied  force  will  be  just  sufficient  to  produce  equilibrium 
if  PX  r,  =  IV  X  r. 

As  a  statical  machine — one  in  which  it  is  desired  to 
maintain  a  condition  of  rest  and  equilibrium  —  the 
wheel-and-axle  is  no  more  useful  than  a  simple 
lever.  Even  if  P  were  applied  at  a  point  on  the 
periphery  of  the  wheel  such  that  A,  F  and  B  are  not  in 
the  same  straight  line,  as  shown  in  Fig.  85,  the  result 


MACHINES. 


133 


would  be  the  same  if  we  had  used  a  bent  lever.  The 
wheel-and-axle  and  the  lever  are  thus  associated  with  each 
other,  because,  for  purpose  of  computation,  the  principle  of 
moments  is  readily  applicable  to  both.  But  as  useful 
machines  they  present  this  difference,  that  the  wheel-and- 
axle  permits  of  continuous  motion,  while  the  available 
motion  of  the  ends  of  a  lever  is  limited  to  the  small 
distance  resulting  from  an  angular  change  of  less  than  90° 


W 


Fig.  86. 

around  the  fulcrum.  When  the  lever  has  turned  through  an 
angle  y  (Fig.  86),  if  the  directions  ofP  and  J^have  remained 
unchanged,  their  moments  will  become  P  b  cos  y  and  Wa 
cos  y,  and  are  zero  when  y  =  90°. 

When  the  wheel-and-axle  is  used,  Pand  W  do  not  follow 
the  points  B  and  A  to  a  new  position  (Fig.  87),  as  rotation 
occurs,  but  are  shifted  to  new  points  of  application,  in  such 
manner  that  the  angles  WAF  and  PBF 
(Figs.  84,  85  and  87),  may  be  kept  equal 
to  90° ,  —  the  position  of  maximum 
moment. 


It  is  not  always  convenient,  however, 
to   preserve   this   favorable  direction  of 
P  and  W  with  reference  to  their  lever 
arms.     In   many  contrivances,  like  the 
Fig.  87.  windlass,    the  wheel   is   replaced   by   a 

crank,  which  acting  like  a  single  lever  loses  the  advantages 
of  a  wheel.     The   rope   of  a  windlass   is   always   at   right 


134  THEORETICAL   MECHANICS. 

angles  to  its  instantaneous  lever  arm  in  the  axle,  but  the 
force  applied  to  the  handle  attached  to  the  crank  is  not 
always  exerted  to  the  greatest  advantage,  being  sometimes 
at  right  angles  to  the  crank  and  at  other  times  acting  at  a 
less  favorable  angle.  This  is  especially  true  of  the  crank 
of  an  engine,  upon  which  the  connecting  rod  acts  at  a 
varying  angle.  The  spoke  of  a  capstan  is  analogous  to  the 
crank  of  a  windlass  in  construction,  but  a  person  operating 
a  capstan,  by  walking  in  a  circle,  is  always  pushing  at  right 
angles  to  the  lever  arm,  and  hence  nothing  is  lost  by  not 
having  the  entire  wheel. 

When  a  rope  or  belt  acts  on  the  circumference  of  a 
wheel  an  error  is  introduced  in  computing  the  moment 
unless  we  add  to  the  radius  of  the  wheel  one  half  the 
thickness  of  the  rope  or  belt,  for,  if  the  rope  is  wrapped 
around  the  circumference  sufficiently  to  prevent  slipping,  it 
moves  entirely  with  the  wheel  and  the  force  is  distributed 
equally  over  the  cross-section  of  the  rope  or  belt.  Perhaps 
this  can  be  made  a  little  clearer  by  the  principle  of  work. 
As  long  as  the  rope  is  coiled  around  the  wheel  the  inner 
strands  are  compressed,  and  the  outer  portions  elongated, 
beyond  their  normal  condition.  If  a  force  is  exerted  on  the 
rope  causing  the  wheel  to  turn,  the  distance  through  which 
the  force  moves  is  the  length  of  rope  unwound,  after  the 
compressed  and  elongated  portions  have  returned  to  their 
normal  condition.  Hence,  in  determining  the  work  done 
by  the  force  in  moving  the  wheel  through  any  angle,  we 
take  as  the  distance  through  which  the  force  moves  an  arc 
described  with  a  radius  equal  to  the  radius  of  the  wheel 
plus  half  the  thickness  of  the  rope. 

Exercise : 

Give  Jive  examples  of  the  vuheel-and-axle,  and  if  any  involve  the 
use  of  a  crank,  state  whether  the  force  acting  on  the  crank  maintains 
the  position  of  maximum  moment. 


MACHINES.  135 

Gearing  and  Shafting.  The  various  combinations  of 
shafts  and  pulleys,  used  so  *  commonly  in  the  tramsmission 
of  power,  operate  by  a  propagation  of  motion  and  force  from 
one  wheel-and-axle  to ,  another.  The  action  of  the  entire 
system  is  traced  from  part  to  part  in  the  same  manner  that 
a  series  of  compound  levers  would  be  analyzed  into  its 
component  simple  levers. 


<_'*' 
Fig- 

For  example,  if  a  force  of  P=  90  Ibs.  (Fig.  88)  is  applied 
at  the  circumference  of  a  24-inch  pulley  fastened  on  the 
same  shaft  with  a  9-inch  pulley,  it  will  be  equivalent  to  an 

24. 

opposing  force  of   W  =     -  X  90,  or  240  Ibs.,  applied  to  the 

small  wheel.  Now,  instead  of  moving  the  weight  W, 
suppose  it  is  desired  to  convey  this  action  to  a  second 
wheel-and-axle  and  at  the  same  time  to  modify  it  somewhat. 
If  it  suits  our  needs  we  may  use  a  30-inch  pulley  for  the 
larger  of  these  wheels  and  a  6-inch  pulley  for  the  smaller. 
By  connecting  the  9-inch  pulley  of  the  first  wheel-and-axle 
with  the  3O-inch  pulley  of  the  second  by  means  of  a  belt, 
these  two  wheels  may  be  constrained  to  move  with  the  same 
lineal  velocity,  and  the  force  of  240  Ibs.  at  the  perimeter  of 
the  9-inch  wheel  is  exerted  through  the  belt  and  thus 
applied  to  the  perimeter  of  the  3o-inch  wheel.  This  force 

at  B^  will  balance  a  load  of  Wl  —  ^—  X  240,  or  1 2oolbs.  at  A  l . 
Exercise  :     Verify  this  result  by  the  principle  of  work. 


136 


THEORETICAL  MECHANICS. 


When  the  speed  is  geared  down  to  such  an  extent  that 
very  large  forces  are  involved,  it  might  not  be  practicable  to 
use  leather  belting  in  the  ordinary  manner,  because  the 
belt  would  slip  on  the  pulley  before  such  forces  could  be 
exerted.  If  a  rope  is  used  it  can  be  coiled  around  each 
pulley  as  many  times  as  may  be  necessary  to  make  it  hold 
without  slipping,  (provided,  of  course,  that  the  rope  is  stout 
enough  to  stand  the  tension).  Or,  if  the  shafts  can  be 
placed  close  to  each  other  the  power  can  be  transmitted 
from  one  to  the  other  by  means  of  cog-wheels,  as  shown  in 


Fig.  89. 

Fig.  89, — or  by  means  of  bevel  gears  if  the  shafts  are  not 
parallel.  Since  the  number  of  teeth  in  the  two  cog-wheels 
is  proportional  to  their  circumferences,  the  computed  relation 
between  P  and  W^  will  be  the  same  as  if  we  had  used  the 
belt,  but  the  possibility  of  slipping  is  obviated.  Notice, 
however,  that  the  direction  of  rotation  of  the  second  shaft 
is  opposite  to  what  it  would  have  been  if  the  transmission 
had  been  by  belt. 


MACHINES.  137 

From  this  figure  the  analogy  between  a  system  of 
gearing  and  a  system  of  compound  levers  is  readily  observed. 
If  BFA  were  a  lever,  instead  of  mere  points  in  a  wheel- 
and-axle,  the  downward  force  at  B  would  cause  an  upward 
motion  at  A.  From  A  the  action  would  be  transmitted  to  J5l 
and  the  upward  force  on  jBl  would  have  a  lever  arm 
B,  F,  ;  etc. 

This  holds  true  whether  the  transmission  is  by  belting 
or  by  spur  gearing.  The  only  probable  source  of  error  in 
either  case  would  be  in  misunderstanding  the  real  relation 
between  the  3O-inch  and  g-inch  wheels;  although  these 
pulleys  may  be  of  different  diameters,  they  do  not  give  rise 
to  the  same  mechanical  considerations  as  if  they  were  on  the 
same  shaft.  Two  pulleys  fixed  on  the  same  shaft  constitute 
a  wheel-and-axle,  in  which  case  they  would  have  the  same 
angular  velocities,  and  hence  would  have  different  lineal 
velocities  for  points  on  the  perimeters.  But  being  mounted 
on  different  shafts  and  one  taking  motion  from  the  other  at 
their  perimeters,  as  in  the  instance  under  consideration, 
they  have  the  same  lineal  velocity,  (but  different  rates  of 
rotation).  Therefore,  in  accordance  with  the  principle  of 
work,  the  action  is  propagated  from  the  9-inch  wheel  to  the 
3O-inch  wheel  without  gain  or  loss  of  force. 

THE  PULLEY.  In  machine  shop  practice  the  word 
1 '  pulley  ' '  is  employed  to  designate  a  wheel  over  which  a 
belt  runs  in  a  system  of  shafting,  as  explained  under  the 
wheel-and-axle.  I/ong  before  the  transmission  of  power  by 
belts  and  ropes,  the  wheel  was  used  in  a  somewhat  different 
manner  as  one  of  the  simple  mechanical  powers,  and  it  is 
in  connection  with  this  manner  of  usage  that  we  apply  the 
generic  term  The  Pulley.  Sometimes  the  mechanical 
advantage  of  the  Pulley  consists  only  in  a  change  of 
direction, — a  force  applied  in  some  convenient  direction 


138 


THEORETICAL   MECHANICS. 


being  employed  to  overcome  an  equal  force  acting  in  any 
other  direction;  under  other  circumstances,  by  a  different 
arrangement  of  conditions,  it  may  be  made  to  modify  the 
force  or  motion.  Or,  it  may  afford  both  of  these  advantages, 
especially  when  several  wheels  are  compounded  in  the  same 
machine. 

Fixed  Pulley.  Very  frequently  a  single  pulley,  arranged 
in  the  manner  shown  in  Fig.  90,  is  used  for  purposes 
of  hoisting.  A  grooved  wheel  is  pivoted  in  a  framework, 
which  is  suspended  from  a  rigid  support.  In  the  matter  of 
equilibrium  between  P  and  W,  friction  in  this  instance  is  an 
important  consideration.  But  disregarding  this,  it  is 


Fig.  90.  Fig.  91. 

obvious   that   according   to   the   principle  of  virtual   work 
P  =  W. 

This  is  still  true  if  the  applied  force  is  exerted  in  any 
direction,  other  than  vertical ;  or  even  if  the  free  end  of 
the  rope  is  carried  by  means  of  other  fixed  pulleys  to  any 
distance  where  it  may  be  convenient  to  apply  the  action. 

Exercise : 

In  the  case  of  the  fixed  pulley  referred  to  in  Fig.  go,  in  which  both 
P  and  W  act  vertically  downward,  what  total  load  is  the  rigid 
support  required  to  sustain  f 


MACHINES.  139 

Movable  Pulley.  If  a  single  pulley  is  arranged  in  such 
manner  that  one  end  of  the  rope  is  fixed  to  the  support, 
while  the  pulley  itself  is  free  to  move  with  the  weight, 
the  relation  between  P  and  W  is  greatly  changed.  Also, 
in  this  case,  allowance  has  to  be  made  for  the  weight  of  the 
pulley  as  well  as  for  friction.  But  assuming  a  weightless 

W 
pulley,  P  will  be  equal  to  — ,  because  if  motion  takes  place 

the  free  end  of  the  rope  will  move  twice  as  fast  as  the 
weight.  To  make  this  clear,  imagine  that  the  entire  pulley 
and  weight  were  grasped  in  the  hand  and  raised  to  a  height 
of  one  foot.  The  rope  would  then  be  left  dangling  through 
a  vertical  length  of  one  foot  on  each  side,  and  to  take  up  the 
slack,  the  free  end,  to  which  P  is  applied,  would  have  to  be 
raised  two  feet. 

Ivooked  at  as  a  simple  question  of  statics,  it  will  be 
observed  that  the  total  weight  is  supported  equally  by  the 
two  ends  of  the  rope. 

If  the  two  parts  of  the  rope  are  not  parallel,  the  relation 

W 
P—  — i  just  deduced  for  the  single  movable  pulley,  is  no 

longer   true.      If  force   is   applied    in    a    direction   that    is 
not  parallel  with  the  direction  of   Wy  the  pulley  and  weight 
will  not  remain  vertically  below   the 
point  of  support,  but  will  roll  to  one  side 
until  they  reach  a  position  where  the 
two  parts  of  the  rope  make  equal  angles 
with  a  vertical  line  through  the  center 
of  gravity  of  the   pulley  and  weight 
(Fig.  92).     The  two  parts  of  the  rope 
92.  will  still  be  under  equal  tensions,  and 

the  load  /\>  on  the  point  of  support, 
in  the  direction  of  the  rope  on  that  side,  will  still  be  equal  to 


1 4o 


THEORETICAL   MECHANICS. 


W 
the  applied  force  P,   but  P  will  not  be  equal  to    — .     We 

must  treat  Pa  and  Pb  as  two  component  forces,  and  to  pro- 
duce equilibrium,  their  resultant  must  be  equal 
and  opposite  to   the  known  force    W.     Hence 
knowing  the  values  of  W  and  ft,  and  knowing 
that  the  two  components  Pa  and  /\>  are  equal, 
we  can  compute  the  value  of  the  applied  force 
from    the    relations     of     the     parallelogram 
If  R   represents   this   resultant,    equal   to    W, 
R  =  2  Pa  cos  ft; 
W 


Fig.  93- 

(Fig.  93) 
then 


whence 


Combinations  of  Pulleys.  There  are  two  ways,  illus- 
trated in  Figs.  94  and  95,  by  which  a  fixed  pulley  and  a 
movable  one  may  be  used  in  combination.  Later  on  it 
will  be  shown  that  these  combinations  are  not  essentially 
different,  but  for  the  present  we  will  consider  them  separately. 


MACHINES. 


141 


In  the  first  case  only  one  end  of  the  rope  moves  when  the 
weight  is  raised,  the  other  end  being  attached  to  the  fixed 
pulley.  In  the  other  case  every  part  of  the  rope  moves 
when  the  machine  is  in  operation. 

In  the  first  case  the  applied  force  would  move  twice 
as  fast  as  the  weight  and  hence  P  -  H7/2.  For  "gain- 
ing power ' '  this  arrangement  of  the  two  pulleys  has  no 
advantage  over  the  single  movable  pulley,  but  it  has  the 
possible  advantage  that  by  passing  the  rope  over  the  fixed 
pulley,  the  force  is  exerted  downward  instead  of  upward. 
This  difference  is  shown  in  Figs.  96  and  97,  placed  side  by 
side.  If  the  rope  were  not  passed  over  the  upper  pulley,  as 
shown  by  the  dotted  line,  this  pulley  would  serve  no  purpose 
whatever  ;  and  even  when  it  is  called  into  use  the  free  end 
of  the  rope  does  not  move  any  faster  for  that  reason,  and 


Fig.  96.  Fig.  97. 

there  is  no  further  gain  of  power, — which  is  true,   as  we 
have  learned,  of  all  fixed  pulleys.     The  weight  drags  down 


142  THEORETICAL   MECHANICS. 

on  two  ropes,  each  of  which  supports  IV/2,  and  the  free  end 
of  the  rope  merely  balances  one  of  these. 

In  the  second  case  (Fig.  95),  where  one  end  of  the  rope  is 
attached  to  the  movable  pulley,  P  -  IV/3,  as  may  be  readily 
proved  by  the  principle  of  work.  Statically,  this  follows 
from  the  fact  that  the  weight  is  supported  equally  by  the 
three  ropes. 

In  these  two  cases  the  load  sustained  by  the  point  of 
support  is  very  different.  In  the  first  case  it  is  3/2  W  ;  in 
the  second  case  it  is  2/3  W.  Because,  in  the  first  case  there 
are  three  ropes  dragging  downward  and  each  is  under  a 
tension  of  IVJ2\  while  in  the  second  case  there  are  only  two 
ropes,  whose  tensions,  each  iVj^  exert  a  downward  force 
upon  the  support.  In  each  system  if  we  replace  the  weight 
by  a  rigid  support,  the  pull  will  be  3  P  on  one  support  and 
2  P  on  the  other.  In  other  words,  the  second  system  is  the 
same  as  the  first,  reversed  end  for  end. 

From  the  relations  just  deduced  it  will  be  observed,  as  a 
general  principle  of  pulleys,  that  if  we  disregard  friction, 
etc.,  the  tension  is  the  same  in  all  parts  of  the  rope*;  and 
this  tension  is  equal  to  the  quotient  of  total  load  at  either 
end  of  the  system  —  either  the  weight  or  the  load  sustained 
at  the  point  of  support  —  divided  by  the  number  of  parts  of 
the  rope  at  that  end.  For  example,  in  the  first  of  the  two 
cases  just  considered,  the  load  on  the  support  was  3/2  W,  in 
the  same  ratio  as  the  number  of  ropes  3:2;  and  the 
tension  throughout  the  rope  is  W-±-2,  or  3/2  W-^-^  the 
former  being  the  quotient  for  one  end  of  the  system,  and 
the  latter  being  the  quotient  for  the  other  end.  In  the 
second  case  the  load  on  the  supporting  point  was  2/3  IV,  the 
number  of  ropes  at  the  two  ends  of  the  system  being  in 
the  same  ratio  2:3,  and  the  tension  in  the  rope  being 


This  does  not  apply  to  the  different  ropes  used  in  the  systems  illustrated  in 
Figs.  99,  100  and  101. 


MACHINES.  143 

From  this  consideration  it  follows,  as  we  have  already 
asserted,  that  there  is  no  essential  difference  between  these 
two  cases  ;  when  a  pull  is  exerted  on  the  free  end  of  the 
rope,  a  force  is  called  into  action  at  each  end  of  the  system 
of  pulleys.  We  call  the  ' '  movable  pulley  ' '  the  one  that 
moves  first.  The  fact  that  the  other  end  did  not  move,  or 
that  the  two  ends  did  not  move  together,  is  a  circumstance 
entirely  independent  of  any  consideration  essential  to  the 
pulley.  Since  pulleys  are  used  to  overcome  some  resistance 


W////////////////////A 


Fig.  98.  Fig.  99. 

by  fastening  the  other  end  of  the  system  to  a  fixed  support, 
the  movable  end  is  always  the  one  at  which  the  "weight" 
W,  is  assumed  to  be  placed. 

Exercises : 

1.  A  system  of  four  pulleys  is  arranged  in  the  manner  shown  in 
Fig.  98,  and  the  two  ends  are  connected  with  spring  balances.  If  a 
pull  of  2  Ibs.  is  exerted  at  the  free  end  of  the  rope,  what  will  be  the 
reading  of  each  of  the  spring  balances  ? 

Disregard  friction,  weight  of  pulleys,  etc.  In  practice,  the  pulleys 
would  be  of  the  same  size  and  would  be  placed  side  by  side  in  each 
"block."  They  are  shown  differently  in  the  diagram  for  the  sake  of 
clearness. 


144 


THEORETICAL   MECHANICS. 


2.  When  the  upper  balance  reads  100,  what  is  the  magnitude  of 
the  applied  force,  and  what  is  the  reading  of  the  other  balance  f 

3.  Find  the  relation  between  P  and  W,  the  tension  in  each  rope, 
and  the  load  on  supporting  beam,  for  the  system  of  pulleys  shown  in 
Fig.  Q9. 


Fig.  loi. 


4.  Find  the  relation  of  P  and  W,  the  tension  in  each  rope,  etc. , 
for  the  system  shown  in  Fig.  100. 

5.  Find  the  relation  between  P  and  W,  tensions  in  ropes,  etc.t 
for  the  system  shown  in  Fig.  101. 


THE  INCLINED  PLANE.  As  applied  to  tools  and  machines 
the  Inclined  Plane  is  represented  in  chisels  and  other 
edge-tools,  nails,  screws,  wedges,  cams,  eccentrics,  propeller 
blades,  etc.  The  name  is  derived  from  the  primitive  device 
of  a  plane  surface  used  to  elevate  a  ' '  dead  load  ' '  to  some 
desired  altitude  by  means  of  any  "easy"  incline — a  skid, 
for  example. 


MACHINES. 


The  steepness  of  a  gradient  or  slope  is  sometimes 
designated  by  the  number  of  degrees  in  the  angle  between 
the  inclined  plane  and  the  horizon,  but  it  is  more  commonly 
expressed  as  a  percentage.  This  percentage  is  the  quotient  of 
R  C 
^j-g,  (Fig.  102) — the  altitude  and  base  of  a  right  triangle,  of 

which  the  hypotenuse  is  any  part  of  the  inclined  surface. 
c 

W 
~/\, 


Fig.  102. 


Fig.  103. 


The  mechanical  advantage  of  the  inclined  plane  depends 
upon  the  steepness  of  the  gradient  and  the  direction  in 
which  the  applied  force  is  impressed.  Let  the  right  triangle 
ABC  (Fig.  103)  represent  an  inclined  plane,  of  which  AB 
is  the  horizontal  base.  I/et  the  base,  altitude,  and  hy- 
potenuse be  represented  by  &,  h  and  /,  respectively.  To 
prevent  the  weight  W  from  sliding  down  the  plane  let  a 
force  be  applied  in  a  direction  parallel  to  the  slope,  or 
hypotenuse,  AC.  If  there  were  no  friction  what  would  be 
the  relation  between  P  and  W  to  produce"  equilibrium? 
This  problem  can  be  solved  by  either  of  two  methods  :  (i) 
by  the  principle  of  work,  or  (2)  by  resolving  the  gravita- 
tional action,  W,  into  two  components  parallel  and  per- 
pendicular to  the  plane. 

i.     By  the  Principle  of  Work. 

Since  gravity  acts  vertically,  no  work  is  done  in  moving 
a  body  horizontally  with  a  uniform  velocity,  (unless  there  is 
friction).  Hence,  the  work  done  against  gravity  in  moving 


146 


THEORETICAL   MECHANICS. 


W  from  A  to  C  is  Wh — the  same  as  if  it  had  been  raised 
vertically  from  B  to  C.  The  work  done  by  P  is  measured 
by  the  distance  moved  in  the  direction  in  which  it  is  applied, 
and  is  equal  to  PI. 

By  the  principle  of  work  PI  -    Wh, 

p-*w. 


2. 


By  Resolution  of  the  Vertical  Force,   W. 

If  we   resolve    W  into    two    components,   wl  and  w2, 
(Fig.  104),  one  perpendicular  and  the  other  parallel  to  AC, 


Fig.  104. 

the  former  will  have  no  tendency  to  move  the  weight  one 
way  or  the  other  on  the  plane.     The  latter  component,  w2r 
is  the  only  part  of  W^that  P  is  required  to  equilibrate. 
Hence,  to  produce  equilibrium 

P  =  w2  =   £Fsin  ft. 
From  the  triangle  ABC, 

sin  ft  =  —j-,     whence 

P  —  -—  W,  as  already  proved. 


MACHINES. 


147 


When  P  is  not  Parallel  to  the  Plane.  When  P  is  applied 
in  a  direction  making  an  angle  y  with  the  hypotenuse,  as 
shown  in  Figs.  105  and  106,  it  is  only  partially  effective  in 
the  direction  AC,  the  component  pl  perpendicular  to  AC 
being  entirely  useless  in  any  direction  at  right  angles  to  itself, 
(except  to  alter  the  amount  of  friction  between  W  and  the 
plane,  which  we  are  now  disregarding).  If  />2  is  the 
component  of  P  parallel  to  A  C,  then  by  the  last  paragraph 


-     -. 
cos  y 


But/>=  /^cosy,   whence  Pcosy  =  W  sin  ft,  orP—W 


W 


Fig.  105. 


Fig.  106. 


A  very  important  ease  arises  when  the  direction  of  P 
is  parallel  to  AB.  The  angle  y  is  then  equal  to  ft.  Using 
the  expression 


P  =    W 


sin  ft 
cos  y' 


if  y  =  J8)  it  becomes  P  -  W 


sin  ft 
cos  /8 


=  IV  tan  ft. 


Referring  to  the  triangle  it  will  be  seen  that 

tan  ft  =  — -,  whence  P=-rW. 
o  o 

That  is,  if  the  power  is  applied  parallel  to  the  base,  P  and 
W  are  in  the  inverse  ratio  of  the  base  and  height.     This, 


148 


THEORETICAL   MECHANICS. 


of  course,  is  verified  by  the  principle  of  work  ;  for  when  the 
weight  is  moved  up  the  plane  from  A  to  C  it  moves  through 
the  horizontal  distance  b,  and  the  vertical  distance  h;  and 
P  being  applied  horizontally  does  an  amount  of  work  equal 
to  Pb,  which  we  know  must  be  equal  to  Wh.  If  Pb  —  Wh, 
then 

P=~W 

o 

The  Wedge.  It  has  just  been  shown  that  when  the 
"weight"  on  an  inclined  plane  acts  in  a  direction  per- 
pendicular to  the  base,  while  the  power  is  applied  parallel 

to  the  base,  the  relation  between  P  and    W  is  P  -  .-—  W. 

b 

A  practical  illustration  of  this  would  be  the  device  shown 
in  Fig.  107.  A  rod  fitted  into  the  guides  G  and  Glt  and 


Fig.  107. 

hence  free  to  move  only  in  a  vertical  direction,  is  held  down 
against  an  inclined  plane  by  means  of  a  weight  W.  A  pres- 
sure P  is  applied  to  the  back  of  the  inclined  plane,  which  is 
pushed,  wedge-like,  under  the  rod.  If  the  first  position  is 
such  that  point  A  is  vertically  under  the  rod,  and  the  plane 
is  pushed  through  its  entire  length,  until  C  comes  under  the 


MACHINES. 


149 


rod,  the  latter  will   have   been   raised   through    a   vertical 
distance  BC,  while  P  moves  through  the  horizontal  distance 

BA.     By  the  principle  of  work  Pb  =    Wh,  or  P  =  —  W. 

It  happens,  however,  in  devices  where  the  wedge  is 
used,  that  W\&  not  always,  or  even  usually,  vertical, — that 
is,  perpendicular  to  the  base  of  the  plane.  Very  frequently 
it  is  perpendicular  to  the  hypotenuse,  as  in  the  case  of  a 
carpenter's  chisel  used  to  split  a  piece  of  wood  in  the 
manner  shown  in  Fig.  108.  If  the  angle  at  the  edge  of 


Fig.  109. 


Fig.  1 08. 

the  chisel  is  ft,  and  the  cohesion  of  the  wood  causes  a 
pressure  W  at  right  angles  to  the  slope,  then  the  conditions 
of  the  problem  are  as  represented  in  Fig.  109.  W  is 
resolved  into  components  perpendicular  and  parallel  to  AB, 
and  the  problem  is  solved  by  statics  or  by  the  principle  of 
work. 


150 


THEORETICAL  MECHANICS. 


Statically,  the  component  w2  is  the  only  part  of  W 
opposing  P  and  tending  to  prevent  the  wedge  of  the  chisel 
from  pushing  into  the  wood,  (for  the  component  w^  can 
exert  no  influence  in  a  direction  at  right  angles  to  itself). 
And  w2  =  I^sin  ft.  To  produce  equilibrium  P  =  w2t  or 
P  =  ^sin  0. 

Or.  by  the  principle  of  work,  if  the  wedge  of  the 
chisel  is  pushed  into  the  wood  from  A  to  C,  the  spread- 
ing of  the  wood  through  a  distance  BC  will  signify  that  the 
component  wl  is  overcome  through  that  distance,  requiring 
an  amount  of  work,  or  expenditure  of  energy,  equal  to 
7v l  X  h.  (The  component  wz  is  not  now  considered,  for  the 


Fig.  i 10 

reason  that  its  point  of  application  is  not  moved  through 
any  distance  in  its  own  direction, — that  is,  parallel  to  AB). 
Since  the  force  P,  parallel  to  the  base,  does  this  work  by 
moving  through  the  distance  BA,  or  b,  we  have  Pb  -  w^  h, 

h 


or 


But 


P  - 

t> 

-r-  -  tan  /3,  and  wv   -    W  CQS 
b 


MACHINES.  151 

Whence  by  substitution,  P  =    J^sin  ft,  as  already  proved  by 
the  static  method. 

In  most  cutting  tools  the  edge  has  the  V  form  of  the 
typical  wedge,  illustrated  in  Fig.  no.  A  wedge  of  this  sort  is 
obviously  equivalent  to  two  inclined  planes,  having  a  com- 
mon base — as  represented  by  the  vertical  dotted  line.  By 
driving  the  wedge  full  length  into  the  wood,  the  fibres  are 
spread  twice  as  far  as  if  the  wedge  had  been  a  right  triangle 
instead  of  isosceles.  The  resistance  W  is  taken  as  being 
perpendicular  to  each  face.  Therefore,  the  relation  between 
P  and  W  is  P  -  2  W  sin  /3.  The  same  relation  is  sometimes 
expressed  in  terms  of  the  total  angle  at  the  vertex  of  the 

" 

wedge,  or  P-  2 


In  the  ordinary  use  of  a  wedge  for  rough  work — such,  for 
instance,  as  we  have  selected  for  our  illustration — the  amount  of 
friction  is  very  great.  But  in  cases  where  accurate  calculations  would 
be  at  all  necessary  the  friction  would  be  reduced  to  a  reasonable 
limit,  where  it  could  be  taken  into  consideration  and  properly 
allowed  for  in  the  computations. 

The  results  of  the  preceding  computations  for  the  Inclined 
Plane  and  Wedge  are  recapitulated  in  the  following  table,  showing 
the  relations  between  P  and  Wfor  the  different  directions  in  which 
these  forces  are  usually  applied  : 


(  W  perpendicular  to  base         (  p  =  _h_  ^  =  IV  Bin  ft 

\  P  parallel  to   hypotenuse    i  / 

f  W  perpendicular    to    basej^  =j?Lw= 

{  P  parallel  to  base  b 

t  W  perpendicular  to  hypotenuse    \  p  =  _h_  w= 

i  P  parallel  to  base  »  / 

/  For  "  F"  wedge  ^  ^ 

•j  W  perpendicular  to  sides  I P  =  2  —j-  W  —  2  £Fsin  j8 

(  P  perpendicular  to  back  ) 


152 


THEORETICAL    MECHANICS. 


The  Screw.  The  Screw  is  a  combination  of  the  Inclined 
Plane  (in  a  modified  form)  and  the  Wheel-and-Axle.  The 
wheel-and-axle  element  is  illustrated  by  taking  a  forged 
bolt  before  the  thread  is  cut  and  applying  a  wrench  to  the 
head.  The  wrench  will  turn  in  a  large  circle,  constituting 
the  "  wheel,"  and  the  shank  of  the  bolt  will  be  the  "axle." 
Cutting  the  thread  adds  the  element  of  the  inclined  plane, 
as  may  be  shown  by  the  following  simple  experiment.  Cut 
a  piece  of  paper  into  triangular  form,  to  illustrate  an 
inclined  plane.  The  grade  or  slope  of  this  plane  will 
depend  upon  the  diameter  of  the  bolt  and  the  desired  pitch 
of  the  screw.  Place  the  edge  BCoi  the  plane  against  the 
side  of  the  bolt  and  parallel  to  the  axis  of  the  cylinder. 
By  wrapping  the  paper  continuously  around  the  bolt,  the 
upper  edge  CA,  representing  the  inclined  surface,  will 
describe  a  helix,  which  will  coincide  with  the  path  along 
which  a  thread  could  be  cut  of  the  desired  pitch. 


Fig.  in. 


The  pitch  of  a  screw  could  be  defined  in  the  same 
manner  as  an  ordinary  inclined  plane — as  an  angle  measured 
in  degrees,  or  as  a  percentage — but  it  is  easier  to  make 
computations  directly  from  the  number  of  threads  per  inch, 
measured  parallel  to  the  axis.  A  machine  screw  is 
measured  by  two  numbers  (besides  the  length),  one  referring 


MACHINES. 


153 


to  the  diameter  of  the  shank  and  the  other  designating  the 
number  of  threads  per  inch.  A  number  16  pitch  means  16 
threads  per  inch. 

The  form  of  a  thread  is  sometimes  rectangular,  some- 
times F-shaped,  and  sometimes  a  modified  form  with 
rounded  edge  or  vertex.  But  the  shape  or  angle  of  the 
thread  does  not  alter  the  angle  of  pitch. 


55° 


English  or  Whitworth 
Thread. 


Square  Thread 


V-Thread 


60° 


American,  or  Sellers 
Thread 


Modified  Square 
Thread 

Fig.  112. 


Trapezoidal 
Thread 


The  operation  of  a  machine  screw  or  a  bolt  requires  a 
nut.  Both  bolt  and  nut  may  be  movable,  or  one  of  them 
may  be  fixed, — according  to  needs.  A  wood  screw  is  always 
used  in  soft  material,  which  is  compressed  into  necessary 
grooves  by  the  thread  of  the  advancing  screw. 

When  a  force  is  properly  applied  to  the  screw,  the  action 
of  the  thread  is  analogous  to  that  case  of  the  inclined  plane 
where  P  is  applied  to  the  back  of  the  plane  and  W  acts 
perpendicular  to  the  base.  There  is  the  additional  con- 
sideration of  course,  of  the  wheel-and-axle.  It  is  not 
necessary,  however,  to  follow  out  all  these  relations,  because 


154  THEORETICAL    MECHANICS. 

the  total  mechanical  advantage  of  a  screw  can  be  computed 
directly  from  its  pitch  by  the  principle  of  work.     Suppose, 
for  instance,   that  a  bolt    of  ^-inch  diameter,  and  number 
8  pitch,  is  turned  by  means  of  a  wrench  applied  to  the  head. 
The   resistance    W,  let  us  assume,  is  occasioned  by  com- 
pressing two  pieces  of  wood  held  together  between  the  nut 
and    the    head    of    the   bolt.       If    a 
pressure  P  is  exerted  on  the  wrench 
at  a  point  9  inches  from  the  axis  of 
the  bolt,  what  total  resistance  W  can 
it  overcome  ?      One  complete  turn  of 
Fig.  113.  the    wrench   will    cause    the    bolt   to 

advance  through  the  nut  }i  inch  (the 

distance  between  two  adjacent  threads),   or  ^  foot.      The 
distance  moved  by  the  point  at  which  P  is  applied  will  be 


feet. 


12 

Therefore 


12.  8  12 

In  general,  if  h  is  the  distance  between  two  adjacent 
threads,  and  r  is  the  distance  from  the  axis  of  the  screw 
to  the  point  of  application  of  P,  the  relation  between  P  and 
W  would  be  expressed  by  the  formula 

P  X  2irr  =  Wh. 


The  two  distances  r  and  h  must  be  expressed  in  the  same  units, — 
both  in  inches  or  both  in  feet.  In  the  problem  taken  as  an 
illustration  these  distances  were  reduced  to  feet  in  order  to  get  the 
foot-pound  as  a  unit  of  work. 


MACHINES. 


155 


Notice  that  W  is  supposed  to  act  parallel  to  the  axis  of  the 
screw, — which  would  be  perpendicular  to  the  base  of  the  inclined 
plane. 

Notice,  also,  that  the  diameter  of  the  bolt  is  not  used  in  the 
calculations.  It  is  duly  involved,  however,  in  the  pitch,  but  was 
eliminated  from  the  computations  when  we  agreed  to  deduce  the 
mechanical  advantage  directly  from  the  pitch.  The  pitch,  h,  is  equal 
to  TT  X  d  X  tan  j3,  where  d  is  the  diameter  of  the  bolt  and  (3  is  the 
slope  of  the  thread  measured  in  degrees.  Bach  thread  is  equivalent 
to  an  inclined  plane  of  height,  h;  base,7r<aT;  and  slope  |8,  wrapped 
around  the  bolt. 


Endless  Screw.  Consider  a  threaded  cylinder  fitted  into 
guides  in  the  manner  shown  in  Fig.  114.  The  screw  itself 
cannot  advance,  being  restrained  by  the  guides,  but  the 


Fig.  114. 


toothed  wheel  D,  can  be  made  to  rotate  continuously  by 
turning  the  screw.  The  teeth  on  the  circumference  of  the 
wheel  feed  into  the  threads  of  the  screw  from  one  side  and 
out  at  the  other.  From  the  wheel  D  power  may  be  taken, 
after  the  manner  of  the  wheel-and-axle,  and  the  action  is 
endless,  notwithstanding  the  limited  length  of  the  screw. 


156 


THEORETICAL    MECHANICS. 


The  Cam  and  Eccentric.  If  a  circular  disc  is  pivoted 
at  a  point  other  than  the  center,  as  the  point  O  (Fig.  115),  and 
is  made  to  rotate  around  that  point 
it  is  said  to  have  an  eccentric  motion, 
and  is  itself  called  an  eccentric.  If  a 
rod  were  set  in  guides  so  as  to  rest 
vertically  upon  point  S,  it  would  be 
pushed,  up  vertically  from  6"  to  T'  by 
a  semi-rotation  of  the  eccentric.  If 
the  eccentric  is  made  to  rotate  con- 
tinuously, the  rod  will  move  up  and 
down  with  a  reciprocating  motion 
between  positions  5*  and  T'  * 

The  total  work  done  in  raising  the 
Fig.  115.  rod  from  6*  to  T'  is  independent  of  the 

manner   in  which  it  is  accomplished, 

but  it  is  not  proportioned  uniformly  throughout  the  stroke. 
It  is  equivalent  to  pushing  a  weight  up  an  inclined  surface 
of  varying  slope, — or  rather  pushing  such  a  surface  under 
the  weight.  Suppose,  for  instance,  that  six  equal  angles  be 
constructed  around  O  by  means  of  lines  Oa,  Ob,  Oc,  etc. 
As  rotation  takes  place,  the  points  a,  d,  c,  dy  e  and  T  assume 
positions  a',  b\  c',  etc.,  successively.  During  the  first  12 
rotation  the  rod  is  raised  through  the  distance  Sa',  equal  to 
the  difference  between  Oa  and  OS ;  during  the  next  interval 
of  T2  rotation  the  rod  is  raised  from  a'  to  b{ ',  the  difference 
between  Ob  and  Oa,  etc.  Since  a'  b'  is  greater  than  Sa\  it  is 


*The  eccentric  is  used  upon  engines  to  produce  and  control  the  valve  motion, 
but  the  motion  thus  given  to  the  eccentric  rod  is  not  the  same  as  that  given  to  the 
rod  mentioned  in  the  above  illustration.  A  valve  eccentric,  running  in  an 
eccentric-strap,  gives  a  motion  that  is  the  same  as  if  a  crank  of  the  same  swing 
had  been  used  instead  of  the  eccentric. 


MACHINES. 


157 


obvious  that  the  average  slope  of  the  surface  pushed  under 
the  rod  during  the  second  interval  was  greater  than  during 
the  first  interval. 

The  distance  ST'  is  called  the  "swing"  or  "throw"  of 
the  eccentric. 

A  earn  is  essentially  the  same  as  an  eccentric,  differing 
mainly  in  the  shape  of  the  periphery.  By  varying  the 
surface  of  the  disc  an  infinite  number  of  straight-line  motions 
can  be  accomplished.  By  means  of  a  certain  heart-shaped 
disc  ( Fig.  1 1 6)  the  rod  can  be  given  a  uniform  velocity.  If  the 
disc  is  not  symmetrical  the  return  stroke  will  be  different 


Fig.  1 16. 


Fig.  117. 


Fig.  118. 


Fig.  119. 


from  the  out  stroke  (Fig.  nyj.  Several  repetitions  of  the 
straight  line  motion  can  be  accomplished  in  a  single  rotation 
of  the  axis,  by  means  of  lugs  or  shoulders  on  the  edge  of 
the  disc  (Figs.  118  and  119).  Sometimes  the  end  of  the 
rod  plays  in  an  irregularly  grooved  collar  (Fig.  120).  All 
these  cases  involve  the  Inclined  Plane  and  can  be  solved  by 
the  principle  of  work  if  the  swing  is  known. 


158 


THEORETICAL   MECHANICS. 


The  end  of  the  rod  that  bears  on  the  cam  is  usually  fitted 
with  a  roller,  for  which  allowance  must  be  made  in  con- 
structing the  cam  to  give  a  required  motion  to  the  rod. 
Very  frequently  the  rod  is  connected  with  a  system  of  levers 
to  still  further  modify  the  motion  (Fig.  121):  By  the 


Fig.  121, 


Fig.  1 20. 


Fig.  122, 


arrangement  shown  in  Fig.  122,  uniform  motion  of  the  shaft 
bearing  the  eccentric  is  converted  into  irregular  motion  of  a 
second  shaft. 


The  Toggle- Joint.  The  Toggle-joint  is  used  in  adjustable 
carriage  tops  ;  in  printing  presses  ;  in  machines  used  for 
stamping  metals,  leather,  wood,  etc.;  and  occasionally  in 
copying  presses  when  the  desired  pressure  is  greater  than 
could  be  obtained  from  a  screw.  Its  mechanical  advantage 
is  enormous. 

Two  bars,  AB  and  AC  are  pivoted  in  the  manner  shown 
in  Fig.  123,  and  the  ends  B  and  C  are  constrained  to  move 
laterally,  or  at  right  angles  to  the  path  of  A.  A  small 


MACHINES. 


159 


pressure  at  A,  applied  as  shown,  will  produce  a  very  great 
outward  pressure  at  B  and  C,  depending  upon  the  magnitude 
of  the  angle  (3. 

The  relation  between  P  and  W\$>  not  fixed,  but  changes 
as  the  angle  of  the  joint  changes,  the  mechanical  advantage 
of  P  over  W  becoming  greater  and  greater  as  the  rods 
flatten  out.  If  a  constant  force  P  is  applied  at  A  it  will 
give  a  certain  greater  pressure,  W,  at  B  and  C,  but  as  the 
joint  gradually  straightens  out,  approaching  the  straight  line 
BC,  the  value  of  W  becomes  greater  and  greater,  though 
the  applied  force  P  has  remained  constant.  As  point  A 
moves  through  the  distance  AD,  the  end  B  moves  through 


the  difference  between  AB  and  ;/?#.  Now  DB  =  AB  cos  ft, 
and  for  small  angles  cos  ft  is  almost  at  its  maximum  value,— 
that  is,  very  nearly  equal  to  one, — whence  the  difference 
between  AB  and  AD  is  very  small.  Bven  when  ft  is  as 
large  as  5  degrees,  we  find  cos  J3=  0.9962,  and  AB  —  BD  — 
.0038  AB.  As  BAC  approaches  a  horizontal  position  as  a 
limit  this  difference  becomes  almost  infinitesimal  in  com- 
parison with  an  appreciable  movement  of  P  in  direction 
AD,  and  therefore,  at  this  instant — just  before  ft  becomes 
zero,  according  to  the  principle  of  work,  W  is  enormously 
greater  than  P. 


i6o 


THEORETICAL   MECHANICS. 


Differential  Motion.  In  the  differential  pulley,  the 
differential  screw,  and  the  differential  wheel-and-axle,  an 
extra  part  is  added  to  the  machine  with  the  sole  function 

of  reducing  the  motion  of  the 
load  and  thereby  increasing  the 
mechanical  advantage  of  the 
machine. 

In  Fig.  124,  illustrating  a 
differential  wheel-and-axle,  P  is 
applied  to  the  circumference  of 
a  wheel  of  diameter  d" .  The 
weight  is  suspended  from  a  pulley 
which  is  itself  supported  by  ropes 
coiled  around  the  cylinders  ^and 
d ' ,  constituting  two  portions  of  a 
solid  or  continuous  axis,  upon 

which  the  wheel  d"  is  fixed.  If  the  force  P  causes  the 
wheel  and  axis  to  rotate,  the  rope  wrapped  around  d'  will 
be  drawn  up,  but  at  the  same  time  the  portion  coiled  upon 
the  cylinder  d  will  be  unwound.  The  rate  at  which  W 
will  be  raised  will  depend  upon  the  difference  between 
the  circumferences  of  the  two  parts  of  the  axis  (though  we 
must  remember  also  that  the  pulley  supporting  the  weight 
is  a  movable  pulley,  for  which  reason  the  weight  rises  only 
half  as  fast  as  the  rope  is  shortened). 

Everything  duly  considered,  it  will  be  found  that 


Fig.  124. 


P  X 


d'  -  ird 


or 


2d 


MACHINES. 


161 


The  chain-hoist  (or  differential  pulley,  as  it  is  called), 
and  the  differential  screw  are  perhaps  the  most  familiar 
devices  in  which  the  idea  of  differential 
motion  is  used.  In  the  differential  pulley,  two 
wheels  of  slightly  different  diameters  are  fixed 
on  the  same  axis,  after  the  manner  of  a  wheel- 
and-axle.  A  third  wheel  supports  the  weight 
in  the  manner  shown  in  Fig.  125.  Instead  of 
a  rope,  as  ordinarily  used  with  pulleys,  an 
endless  chain  passes  from  pulley  d^  to  the 
movable  pulley  (part /^);  thence  to  the  pulley 
d  (part  /£);  thence  downward,  hanging  free 
(part  /);  and  thence  to- the  circumference  of  dl 
(part  m),  thus  completing  the  circuit. 

Now  if  P  is  applied  to  part  m  of  the  chain, 
the  fixed  pulleys  d  and  dl  will  rotate  together; 
part  h  of  the  chain  will  move  upward  as  far  as 
P  moves  downward,  and  at  the  same  time 
part  k  is  fed  downward  from  the  circumference 
of  d.  But  since  d  and  dl  are  fixed  to  each 
other  and  rotate  together,  the  part  h  is  taken 
up  faster  than  part  k  is  fed  downward,  and  hence  in  a  single 
rotation  of  these  two  wheels  W  will  be  raised  a  distance 
equal  to  one- half  the  difference  between  their  circumferences 
and  P  will  move  downward  through  a  distance  equal  to  the 
circumference  of  the  larger.  Therefore 


Fig.  125. 


d,       - 


or 


P  X  *= 


,  or 


162  THEORETICAL    MECHANICS. 

The  two  wheels  or  cylinders  that  contribute  to  the 
differential  motion  can  be  made  so  slightly  different  that  the 
motion  of  W  may  be  made  as  slow  as  we  please,  and  thus  a 
very  small  applied  force  can  be  made  to  lift  an  enormous 
weight,  up  to  the  limit  of  the  strength  of  the  machine. 


Compound  Machines.  Many  hand  tools  represent  only 
one  of  the  different  elementary  machines  enumerated  on 
p.  in,  but  most  mechanical  contrivances  are  more  complex. 
A  compound  machine  is  one  made  up  of  a  number  of  simple 
machines.  The  "Weight"  of  the  first  becomes  the  "Applied 
Force"  for  the  next,  as  in  compound  levers  (p.  117),  and  the 
train  of  wheels  (p.  136), — though  it  is  not  necessary  that 
the  successive  parts  of  the  machines  should  all  be  of  the 
same  kind.  In  moving  a  house  on  rollers,  for  example,  a 
capstan  is  used  in  combination  with  a  compound  system  of 
pulleys. 


Examples : 

1.  If  two  forces  of  60  pounds  each,  acting  on  the  same  point  at  an 
angle  of  60°  with  each  other,  are  exactly  balanced  by  a  third  force 
making  an  angle  of 120°  with  the  two  given  forces,  find  the  magnitude 
of  the  third  force. 

2.  A  straight  uniform  lever  AB,  12  feet  long,  balances  about  a 
Point  in  it  j  feet  from  B,  when  weights  of  9  pounds  and  13  pounds  are 
suspended  at  A  and  B,  respectively.     Find  the  weight  of  the  lever. 

3.  An  iron  bar  6  feet  long  is  free  to  turn  about  a  horizontal  axis 
which  is  4.  feet  from  one  end.     On  top  of  it  is  placed  a  second  bar  4  feet 
long,  but  otherwise  like  the  first  bar,  so  that  the  lengths  of  the  bars  are 
parallel.     The  two  bars  are  now  balanced  upon  the  pivot.     Describe 
the  position  of  the  top  bar. 


MACHINES.  163 

4.  Analyze  the  action  of  a   claw-hammer  in  drawing  a  nail. 
Where  is  the  fulcrum  f     The  "weight"  f     The  "power"  f 

5.  Why  does  the  driver  of  a  heavy  load  take  a  zig-zag  course  in 
climbing  a  hill  ? 

6.  The  drum  of  a  windlass  has  a  diameter  of  JO  inches,  and  the 
crank  has  a   radius  of  18  inches.     What  minimum  force  must  be 
applied  to  the  handle  to  raise  a  load  of  160  pounds  f 

Disregard  friction  and  thickness  of  rope. 

(/)}     What  difference  if  we  allow  a  half  inch  diameter  of  rope  ? 

7.  The  screw  of  a  letter-press  has  a  pitch  of  %  inch,  and  the 
diameter  of  the  wheel  is  10  inches.     What  pressure  will  be  exerted 
upon  the  copying  book  by  forces  of  25  pounds  applied  at  two  different 
points  on  the  circumference  of  the  wheel  ? 

8.  A  two-arm   balance  is  supposed    to    have    arms    of   equal 
length;  otherwise  it  is  false.     A  false  balance  may  appear  to  be  true 
because  the  beam  stands  at  equilibrium  when  there  is  no  load  in  the 
pan,    but   a   little   thought  will   show  that  it  will  not  continue  at 
equilibrium  when  equal  weights  are  added. 

(a)  If  a  dishonest  dealer  were  using  a  false  balance  of  this 
sort,  in  which  arm  would  he  place  the  substance  being  sold? 

(b]  Can  you  think  of  any  way  by  which  to  test  a  balance  to 
determine  whether  it  is  true  or  false  f 

9.  In   what    way  do    metal-cutting   shears    differ    most  from 
ordinary  scissors  ?     Why  ? 

10.  In  the  chain-hoist  described  on  p.  161,  the  two  wheels  d  and 
dr  have  diameters  o/6y2  inches  and  7  inches,  respectively.     What  force 
P  must  be  applied  for  a  load  W  equal  to  600  pounds  f 

11.  A    1 200-pound   anchor  is  hoisted   by  means  of  a  capstan 
having  a  drum  of  18  inches  diameter  and  four  spokes  each  8  feet  long. 


164  THEORETICAL   MECHANICS. 

What  force  mttst  be  exerted  by  each  of  four  sailors  pushing  at  the 
extreme  ends  of  the  spokes  f 

12.  A  plank  lies  with  one  end  projecting  over  a  log.     A  boy 
weighing  100  pounds  walks  out  on  the  projecting  end,  and  when  he 
gets  6  feet  from  the  log,  the  plank  tips.     The  center  of  gravity  of  the 
plank  is  3  feet  from  the  log.     Find  the  weight  of  the  plank. 

13.  A  plank  weighing  200  pounds  lies  with  one  end  projecting 
over  a  log.    A  boy  weighing  100  pounds  walks  out  on  the  projecting 
end,  and  when  he  gets  6  feet  from  the  log  the  plank  tips.     Where  is  the 
center  of  gravity  of  the  plank  ? 

14.  A  house  on  rollers  is  moved  by  means  of  pulleys  and  a 
capstan. 

(a)  If  the  resistance  to  rolling  is  20  tons,  at  what  rate  can  it 
be  moved  by  a  single  horse  working  at  the  rate  of  one  H.  P.  f 

(b)  If  the  drum  of  the  windlass  has  a  diameter  of  20  inches 
and  the  system  of  pulleys  has  two  wheels  in  each  block,  what  force 
must  the  horse  exert  at  a  point  on  the  arm  of  the  capstan  14  feet 

from  the  center  of  the  drum  f      And  at  what  rate  must  the  horse 
walk  in  order  to  move  the  house  at  the  rate  computed  in  part  (a)  ? 

15.  For  the  endless  screw  shown  on  p.  755,  assume  the  following 
dimensions: 

Pitch  of  screw,  i  inch. 

Crank  arm,  12  inches. 

Number  of  teeth  in  wheel  D,  16. 

Diameter  of  shaft  bearing  wheel,  /X  inches. 

What  force  must  be  applied  to  the  crank  handle  to  lift  a  load  0/300 
pounds  applied  to  the  perimeter  of  the  shaft  f 


CHAPTER  VI. 
FRICTION. 

When  one  body  moves  in  contact  with  another,  whether 
by  sliding  or  by  rolling,  or  when  an  object  travels  through 
a  fluid  medium,  as  a  bullet  through  the  air,  we  instinctively 
assume  that  a  frictional  resistance  inevitably  accompanies 
the  motion.  The  laws  of  friction  are  simple  in  statement, 
but  they  are  quite  empirical  and  do  not  always  hold  with 
that  rigid  mathematical  accuracy  which  characterizes  other 
laws  of  mechanics.  This  is  because  the  usual  phenomenon 
of  friction,  simple  as  it  seems,  is  really  an  aggregation  of  a 
great  number  of  less  obvious  phenomena,  or  of  several 
groups  of  such  aggregations,  which  taken  together  give  rise 
to  a  complexity  of  conditions  surpassing  the  possibility  of 
mathematical  analysis.  For  example,  when  one  surface 
slides  over  another  the  frictional  resistance  is  supposed  to 
be  occasioned  mainly  by  the  interlocking  of  an  infinite 
number  of  particles  on  the  two  adjacent  surfaces.  No 
surfaces  are  smooth  enough  not  to  permit  of  friction  ;  two 
' '  perfectly  smooth  surfaces' '  (if  such  were  possible)  when 
placed  in  contact  with  each  other,  and  the  air  excluded, 
would  be  close  enough  together  for  complete  inter-molecular 
action  between  the  two  surfaces,  and  the  two  bodies  would 
cohere  if  of  the  same  material,  (or  adhere  if  of  different 
materials),  so  firmly  that  they  would  be  as  a  single  rigid 
body.  No  doubt  the  friction  between  two  ordinarily  smooth 
surfaces  is  caused  in  part  by  molecular  attractions  as  well  as 
by  the  interlocking  of  physical  particles,  or  particles  having 
appreciable  dimensions. 


l66  THEORETICAL   MECHANICS. 

Still  more  complicated  is  the  frictional  resistance  between 
lubricated  surfaces,  or  of  a  moving  carriage.  The  rolling 
friction  of  the  tires  in  contact  with  the  ground,  the  sliding 
friction  in  the  bearings,  and  the  frictional  resistance  of  the 
air,  (each  in  itself  a  complex  phenomenon),  are  all  included 
in  the  total  tractional  resistance  of  the  carriage. 

It  will  be  convenient  to  consider  in  order  the  laws, 
(a)  of  Sliding  Friction  ;  (b)  of  Rolling  Friction  ;  (c)  of  the 
Friction  of  Ropes  and  Belts  on  Pulleys,  and  (d)  the  Use  of 
Lubricants . 

Sliding  Friction.  For  the  purpose  of  illustrating  the 
laws  of  sliding  friction,  the  following  results  obtained  by 
actual  experiment  may  be  used  : 

An  ordinary  building  brick,  smoothed  on  one  face  by 
means  of  a  grindstone,  was  placed  on  a  planed  cedar  board. 
A  string  passed  around  the  brick  was  attached  to  a  spring 
balance.  By  pulling  continuously  on  the  balance  it 
gradually  reached  a  tension  at  which  the  brick  began  to 
slide.  Restoring  the  brick  to  its  first  position  under  the 
same  conditions  as  before,  it  was  found  to  move  again  at 
about  the  same  reading  of  tension  on  the  balance, — 2.5 
pounds. 

A  block  of  Oregon  pine,  of  quite  different  dimensions 
from  the  brick,  was  placed  on  the  cedar  board  as  nearly  as 
possible  under  the  same  conditions  as  the  brick,  and  was 
found  to  slide  when  the  pull  was  only  i .  i  pounds. 

How  shall  we  account  for  this  difference — 2.5  pounds  to 
slide  the  brick  and  only  i .  i  pounds  to  slide  the  pine  block  ? 
Was  it  due  to  the  difference  in  the  weights  of  the  two  ;  or 
to  the  area  of  the  surface  of  contact ;  or  to  a  difference  in 
the  natures  of  their  surfaces ;  or  to  all  these  causes  in 
varying  degrees? 


FRICTION.  167 

To  answer  these  questions  the  observations  were 
continued  as  follows  : 

(1)  A  second  pine  block  of  exactly  the  same  dimensions 
as  the  brick  was  loaded  with  weights  until  it  had  also  the 
same  weight  as  the  brick,  5.6  pounds.     Placed  on  the  cedar 
board  under  the  same  conditions  as  before,   it  was  found  to 
slide  under  a  pull  of  3.8  pounds. 

Now,  since  the  brick  and  the  block  have  the  same  weight 
and  the  same  area  of  surface  in  contact  with  the  board,  the 
difference  between  the  friction  of  the  brick  and  cedar  (2.5 
pounds)  and  the  friction  of  the  pine  block  and  cedar  (3.8 
pounds)  must  be  due  to  the  fact  that  the  nature  and  condition 
of  the  brick  surface  is  quite  different  from  the  pine  block. 

In  general,  the  friction  between  two  surfaces  tending  to 
prevent  sliding  is  different  for  different  pairs  of  substances. 
It  has  been  found  in  practice  that  friction  is  generally  greater 
between  surfaces  of  the  same  kind,  as  between  steel  and 
steel ;  leather  and  leather  ;  etc.  Hence  the  advantage  of 
using  brass  bearings  for  steel  shafts,  and  of  covering  with 

leather  the  face  of  a  pulley  used  with  leather  belting. 

» 

(2)  The  opposite  side  of  the  brick  had  not  been  ground. 
Placing  the  brick  on  the  cedar  board  with  the  rough  side 
downward,    the   friction   was   found   to  be    2.8  pounds  as 
against  2.5  for  the  smoothed  surface. 

The  sliding  friction  depends  upon  the  roughness  of 
either  or  both  of  the  surfaces  in  contact.  It  is  obvious 
however,  that  this  roughness  cannot  be  measured  mathe- 
matically. 

(3)  One  of  the  long,  narrow  faces  of  the  brick  had  been 
ground  and  the  opposite  face  left  unground.      Placed  with 
the  smooth  edge  on  the  cedar  board  the  friction  was  found 


1 68  THEORETICAL    MECHANICS. 

to  be  about  2.5  pounds,  the  same  as  for  the  broad  side 
similarly  ground.  The  friction  for  the  ground  edge  was  the 
same  as  for  the  rough  side  of  greater  area. 

In  general,  the  friction  between  two  surfaces  is  indepen- 
dent of  the  area  of  contact  This  law  is  true  within  broad 
limits,  but  when  one  of  the  bearing  surfaces  is  so  small  that 
it  digs  or  cuts  into  the  other,  or  when  the  pressure  is  so 
great  that  the  surfaces  are  deformed  or  abrased,  the  law  is 
interfered  with  by  new  considerations,  quite  distinct  from 
frictional  influences. 


The  most  important  law — that  the  friction  between  two 
surfaces  is  proportional  to  the  force  pressing-  them  together— 
is  deduced  from  the  next  observations. 

(4)  The  Oregon  pine  block  used  above  was  loaded  so 
as  to  weigh  4  pounds,  and  again  placed  on  the  cedar  board. 
The  friction  determined  in  the  same  manner  as  before,  was 
found  to  be  2.6  pounds.  By  placing  additional  weights  on 
the  block,  equal  to  a  total  of  8  pounds,  the  friction  was 
increased  to  5.25  pounds,  or  about  twice  as  much  as  before. 
In  the  same  manner,  when  the  weight  or  pressure  was  made 
three  times  as  large  (12  pounds)  the  friction  was  found 
to  have  been  increased  in  almost  the  same  ratio — to  7.9, 
according  to  the  actual  measurement. 

That  is,  for  two  given  surfaces,  the  friction  depends  only 

upon  the  pressure  between  them.     If  F  is  the  friction,  and 

f 
P  is  the  force   pressing  the   surfaces   together,    then  —=•  is 

constant. 

F 
The  ratio  -»-  is  called  the  Coefficient  of  Friction  for  the 

given  surfaces.      This  idea  is  used  so  much  in  practice  that 
the  coefficient  of  friction  is  usually  represented  in  formulae  by 


FRICTION.  169 

a  special  symbol, — generally  the  Greek  letter  <£.  For 
example,  according  to  our  measurements  the  coefficient  of 
friction  between  the  given  surface  of  Oregon  pine  and  cedar 
as  used  in  the  experiment,  would  be 


=  .68  for  the  first  measurement. 


5-6 

0  -        -=  .65  for  the  second  measurement. 

4 

(f>  =  ~-  -=  .66  for  the  third  measurement. 

8 

=  .66  for  the  fourth  measurement. 


12 


Average  <£  =  .66 

When  we  say  that  the  coefficient  of  friction  between  the 
given  surfaces  of  pine  and  cedar  is  .66  we  mean  that 
the  total  friction  between  them  is  always  that  fraction  of  the 
total  pressure. 

If  the  pressure  acts  in  a  direction  not  perpendicular  to 
the  two  surfaces,  as  by  pressing  a  stick  obliquely  against  the 
block  after  the  manner  shown  in  Fig. 
126,  then  it  becomes  necessary  to 
resolve  the  pressure  into  two  com- 
ponents. Only  that  part  which  acts 
in  a  direction  perpendicular  to  the 
surface  of  contact  is  taken  into  account 
in  considering  the  friction  ;  the  com- 
ponent parallel  to  the  surface  of  contact  will  tend  to  push 
the  block  along  the  plane,  but  it  does  not  thereby  change 
the  friction  one  way  or  the  other. 


THEORETICAL   MECHANICS. 


Example : 

In  the  following  table,  containing  the  values  found  in  the 
preceding  experiments,  insert  in  the  column  headed  "0"  the  value  of 
the  coefficient  of  friction  for  each  pair  of  surfaces  named  in  the  same 
horizontal  line. 


SURFACES 

p 

F 

0 

I 

ii 

Brick 
(On  ground  face) 

Cedar  (planed) 

5-6 

2-5 

Oregon  Pine 
(Planed;  with  weights  added) 

«             «< 

5-6 

3.8 

Brick 
(On  ground  edge) 

«            « 

5-6 

2-5 

Cast  Iron 
(Dressed  on  shaper) 

«             « 

8.7 

4-5 

i  <  .             « 

Leather 
(Old  belting;  flesh  side) 

8.7 

1.8 

<  <               <  i 

Leather 

(Same  piece;  hair  side) 

8-7 

i-5 

Oregon  Pine 

Cedar  (planed) 

4 

26 

«             « 

«  1                              (C 

8 

5.25 

«            i  < 

(«                    <  1 

12 

7-9 

In  all  our  illustrations,  for  the  sake  of  simplicity,  we 
have  considered  the  pressure  between  the  surfaces  as  being 
due  to  the  weight  of  an  object.  But  the  laws  of  friction 
apply  to  pressures  of  all  kinds,  as  in  a  brake  operated  by 
levers  or  by  a  spring.  As  a  further  example,  in  dealing 
with  the  friction  in  the  bearings  of  a  line  of  a  shafting  we 
have  to  consider  not  only  the  downward  pressure  due  to  the 


FRICTION.  iyi 

dead  weight  of  the  shafting  and  pulleys,  but  also  the  force 
exerted  by  the  tension  of  the  belt  (which  may  be  in  any 
direction) ;  it  is  the  resultant  of  the  two  that  gives  the  total 
pressure  of  the  shaft  in  the  bearings — that  pressure  from 
which  we  must  compute  the  friction. 

The  use  of  sliding  blocks  for  our  purposes  of  illustration 
may  also  tend  to  still  another  misunderstanding.  We  are 
not  dragging  the  weight  of  the  block  when  we  slide  it  on  a 
horizontal  surface — at  least  not  in  the  sense  that  we  overcome 
the  weight  in  lifting  it.  The  effort  of  dragging  is  not  due 
to  the  weight  of  the  block,  except  as  the  weight  causes 
friction  at  the  surface  of  contact.  It  should  be  remembered 
that  if  there  were  no  friction  and  the  plane  were  perfectly 
horizontal  the  slightest  force  would  cause  the  block  to  move 
and  to  keep  on  faster  and  faster ;  of  course,  the  greater  the 
mass  of  the  block  and  the  smaller  the  force  applied  to  it,  the 
less  rapidly  would  it  gain  velocity,  but,  as  already  stated  on 
p.  75,  if  there  were  no  friction,  any  force,  howsoever  small, 
would  produce  motion  in  any  mass,  howsoever  large,  on  a 
horizontal  plane. 

Static  Friction  and  Kinetic  Friction.  In  any  of  the 
preceding  observations,  if  we  had  watched  the  reading  of 
the  spring  balance  after  the  body  had  commenced  to  slip, 
we  would  have  seen  that  the  pull  necessary  to  keep  the  body 
moving  is  less  than  just  before  slipping  commenced.  In 
other  words,  the  coefficient  of  friction  between  the  two 
surfaces  when  they  are  in  relative  motion  is  less  than  the 
friction  of  rest,  or  the  statical  coefficient.  However,  the 
friction  during  motion,  or  kinetic  friction,  obeys  the  same 
laws  that  we  have  already  deduced  for  statical  friction,  the 
friction  being  simply  less  by  a  certain  amount  for  the  given 
surfaces,  but  still  proportional  to  the  pressure.  This  is  true 
provided  the  velocity  is  not  too  great ;  at  high  speeds  the 


172 


THEORETICAL    MECHANICS. 


coefficient  is  less,    while   at   very   low  speeds   the    statical 
coefficient  and  the  kinetic  coefficient  are  nearly  equal. 

Not  only  is  the  statical  greater  than  the  kinetic  friction, 
but  it  is  also  true  that  the  longer  the  surfaces  have  remained 
in  quiet  contact  the  greater  the  statical  coefficient. 

Friction  Always  a  Resistance.  In  discussing  the  idea 
of  a  force  (p.  60),  it  was  stated  that  friction  is  not  an  active 
agent  capable  of  producing  motion ;  in  fact  it  has  no 
existence  until  called  into  play  by  our  effort  to  make  the  two 
surfaces  slide  in  contact  with  each  other.  Furthermore,  it 
has  no  direction  of  application  such  as  a  force. always  has  ; 
when  a  brick  rests  on  a  horizontal  surface  the  friction  is  the 
same,  whatever  the  direction  in  which  we  slide  the  body. 

Determination  of  the  Coefficient  of  Friction  by  Means 
of  the  Limiting  Angle,  or  Angle  of  Repose.  This  simple 
method  of  measuring  the  coefficient  of  friction  is 
exemplified  in  the  following  experiment: 

A  block  of  iron  weighing  8.7  pounds  was  placed  on  the 
cedar  board  and  the  latter  gradually  inclined  until  the  block 
B 


Fig.  127. 

began  to  slip  down  the  grade.  The  angle  of  elevation,  y, 
at  which  the  slipping  commenced  was  measured  by  applying 
an  ordinary  two-foot  square  in  such  manner  as  to  determine 
the  lengths  OA  and  AB  in  Fig.  127.  These  lengths  OA 
and  OB  were  found  to  be  20  inches  and  ioy9<r  inches 


FRICTION.  173 

A  ft 
respectively.      The  ratio  -z—r-  is  simply  tan  y,  which,  in  this 


AB       10.56 

case  is  tan  y  =  TJ~T^  =  °-53-       By   comparing   this 

20 


value,  0.53,  with  the  coefficient,  0.52,  for  the  same  surfaces 
(as  previously  determined  by  means  of  the  spring  balance), 
it  will  appear  that  the  coefficient  of  friction  is  simply  the 
tangent  of  the  angle  at  which  slipping  commenced. 

That  this  is  true  can  be  readily  shown  by  simple 
geometric  demonstration.  The  weight  W  is  a  force  acting 
vertically  downward,  but  as  the  plane  of  the  board  is  no 
longer  horizontal,  only  a  part  of  W  —  the  component 
perpendicular  to  OB  —  serves  to  press  the  two  surfaces 
together.  Accordingly,  if  W  is  resolved  into  two  com- 
ponents, one  parallel  to  OB  and  the  other  perpendicular  to 
OB,  the  former  (component  a)  tends  to  drag  the  weight 
down  the  plane,  but  cuts  no  figure  in  the  friction,  while 
the  other,  b,  tends  only  to  cause  friction.  If  <£  is  the 
coefficient  of  friction,  the  total  friction  is  b<$>,  and  if  the 
component  a  is  just  sufficient  for  the  body  to  slip  down  the 
plane,  then  obviously  a  *•=?  b  <f>.  But  a  -  -  W  sin  y  and 
b  =  W  cos  y  ;  whence  W  sin  y  ~  <f>  W  cos  y,  or 

W  sin  y 

9  =£  TTT         -  —  tan  y,  as  was  to  be  shown. 
H^cosy 

Examples: 

1.  A   body  resting  on  a  surface  just  begins  to  slide  when  the 
surface  is  inclined  21°  if.     What  is  the  coefficient  of  friction  f 

2.  A  body  placed  on  a  4.0%  grade  has  just  sufficient  inclination 
to  cause  slipping.     What  is  the  coefficient  of  friction  f 

3.  The  coefficient  between  the  Oregon  pine  and  the  cedar  board 
was  0.66.       What  is  the  greatest  angle  at  which  the  board  could  be 
inclined  without  the  block  slipping? 

4.  If  a  body  is  on  an    inclined  plane  the  total  friction  is   less 
than  on  a  horizontal  plane.     Why  ? 


174  THEORETICAL   MECHANICS. 

Work  Done  in  Dragging  a  Body  by  Sliding.  When  a 
body  is  dragged  with  a  uniform  velocity  along  a  horizontal 
surface,  the  only  work  done  is  in  overcoming  the  frictional 
resistance  (See  p.  75).  If  the  body  is  on  an  inclined  plane 
the  total  friction  is  less  than  on  a  horizontal  plane,  but  there 
is  the  additional  consideration  that  extra  work  will  be 
required  if  the  body  is  being  moved  up  the  incline.  If  the 
body  is  being  moved  down  the  grade  a  part  of  the  weight  of 
the  body  then  aids  in  overcoming  the  friction. 

Examples : 

1.  Resolving  the  weight  into  two  components,  a  and  b,  parallel 
and  perpendicular  to  the  plane   (as  in  Fig.  127],  and  calling  the 
coefficient  of  friction  0,   what  will  represent  the  force  necessary  to 
move  the  body  up  the  plane  ?     What  to  move  the  body  down  the  plane  f 
In  the  latter  question  what  is  signified  if  a  is  greater  than  b  0  f    If 
a  =  b  0  what  is  the  value  of  7  f 

2.  A  mass  weighing  200  pounds  rests  on  a  horizontal  surface.    If 
the  coefficient  of  friction  is  0.23  what  force  will  be  required  to  drag  the 
body  along  the  surface  with  a  uniform  velocity  ?     What  horse-power 
is  required  to  drag  this  body  at  the  rate  of  100  yards  a  minute  ? 

3.  A  body  weighing  go  Ibs.  rests  on  a  surface  inclined  at  an  angle 
7  =  74°  j'.      The  co  efficient  of  friction  between  the  two  surfaces  is  o.j/j. 

(a)     What  force  is  necessary  to  drag  the  body  up  the  plane  f 
(b}     What  force  is  necessary  to  drag  it  down  the  plane  ? 

(c]     What  horse-power  will  be  required  in  each  case  to  keep 
the  weight  moving  at  the  rate  of  40  feet  per  second? 

Rolling  Friction.  It  has  already  been  stated  (Kinematics 
p.  32)  that  when  a  wheel  rolls  along  the  ground  that  part 
of  the  wheel  in  contact  with  the  ground  is  alwa}7s  at  rest ; 
as  each  point  comes  down  and  touches  the  ground  that 
point  on  the  wheel  is  for  the  instant  at  rest  relatively  to  the 


FRICTION.  175 

surface  upon  which  the  wheel  rolls.  There  is  no  sliding  of 
the  wheel  bodily  along  the  ground,  and  yet  there  is  friction, 
which  in  time  will  bring  the  wheel  to  rest. 

It  is  this  resistance  that  is  called  rolling  friction.  As 
the  successive  parts  of  the  circumference  come  down  to  the 
ground  it  is  not  difficult  to  conceive  how  the  minute 
projecting  particles  of  the  wheel  collide  with  similar  elements 
of  roughness  on  the  ground,  producing  this  resistance.  The 
rougher  the  surface  the  greater  the  rolling  friction,  as  we 
know  from  the  fact  that  a  ball  will  roll  farther  on  a  smooth 
floor  than  on  a  carpet,  or  on  a  gravel  path.  If  the  surface 
of  the  wheel  is  soft  it  will  be  flattened  and  its  progress 
retarded.  Likewise,  if  the  surface  upon  which  a 
rolling  body  moves  is  such  that  a  depression  is  formed  by 
the  weight  of  the  body,  a  corresponding  ridge  will  also  be 
formed  in  the  path  of  the  object,  so  that  the  energy  of  the 
rolling  body  will  be  used  up  in  deforming  the  surface  and 
constantly  climbing  this  little  ridge. 

If  a  wheel  or  ball  rolls  in  contact  with  a  curved  surface,  the 
"normal  pressure"  at  any  instant  would  refer  to  a  direction  perpen- 
dicular to  a  line  or  plane  tangent  in  common  to  the  two  sufaces  at 
their  point  of  contact  for  the  given  instant. 

Work  Done  in  Dragging  Vehicles.  It  has  already  been 
stated  that  when  a  rolling  vehicle  is  moved,  the  tractional 
resistance  includes  all  friction,  whether  sliding  or  rolling, 
both  in  the  bearings  and  at  the  rims  of  the  wheels.  This 
resistance  is  designated  as  so  many  pounds  to  each  ton  of  load, 
including  the  weight  of  the  vehicle.  For  example,  if  a  car 
weighing  12  tons  carries  a  freight  load  of  6  tons,  and 
requires  a  force  of  250  pounds  to  drag  it  with  a  uniform 
velocity,  then  the  tractional  resistance  is  ^-,  or  131  pounds 
per  ton. 


176  THEORETICAL   MECHANICS. 

Examples : 

1.  A  train  is  made  up  of  12  cars  each  representing  a  total  load 
of  23  tons.     If  the  fractional  resistance  is  10  pounds  per  ton  load, 
what  force  will  be  necessary  to  keep  the  train  moving  with  a  uniform 
velocity  f 

2.  What  horse -power  will  be  required  to  keep  this  train  moving 
at  the  rate  0fjo  miles  an  hour  ? 

3.  If  this  train  comes  to  an  8^0  grade  what  horse-power  would 
be  required  to  carry  the  train  up  the  grade  with  a  velocity  of  4  miles 
an  hour? 

Consider  that  the  friction  is  the  same  on  the  slope  as  on  the  level. 


Anti-Frietion  Wheels  and  Ball  Bearings.  When  a  wheel 
supports  a  vehicle  of  any  sort  in  such  a  manner  as  to  require 
bearings,  the  resistance  encountered  in  moving  the  vehicle 
includes  not  only  the  rolling  friction  at  the  circumference 
but  also  some  sliding  friction  in  the  bearings.  To  get  rid  of 
this  sliding  friction,  or  rather  to  reduce  it  to  a  minimum,  a 
number  of  devices,  such  as  anti-friction  wheels,  roller 
bearings  and  ball  bearings,  are  used  under  varying 
conditions. 

The  manner  of  using  anti-friction  wheels  is  shown  in 
Fig.  128.  Suppose  that  a  weight  W  is  carried  on  the 
circumference  of  a  wheel  C.  The  axle  ^  of  this  wheel, 
instead  of  revolving  in  fixed  bearings,  is  supported  by  the 
rirns  of  two  other  wheels,  Cl  and  C2,  the  axles  of  these, 
however,  being  in  fixed  bearings.  As  5  revolves  C^  and  C.2 
also  revolve,  but  with  a  much  smaller  angular  velocity 
Now  the  total  load  and  the  friction  in  the  bearings  of  Cl  and  C2 
may  even  be  greater  than  would  have  existed  in  the  bearings 
of  5  if  fixed  bearings  had  been  used  at  that  point  instead  of 
the  anti-friction  wheels.  But  the  rate  at  which  the  axles  of 
Cl  and  C2  move  in  their  bearings  is  so  slow  that  very  little 


FRICTION. 


177 


work  is  done  during  each  revolution  of  5,  and  hence  it 
happens  that  the  energy  consumed  by  sliding  friction  in 
the  bearings  of  Cl  and  C2  plus  the  rolling  friction  of  5  on 
the  rims  of  Cl  and  Cz  is  less  than  what  would  have  been 
used  up  by  the  sliding  friction  of  5  placed  directly  in  fixed 
bearings. 


Fig.  128. 

Ball  Bearings  have  become  so  well  known  through  their 
extensive  application  to  bicycles  that  the  illustration  in 
Fig.  129  needs  no  special  description.  If  the  axle  5 
revolves  in  the  direction  of  the  hands  of  a  clock,  the  balls 
will  all  revolve  in  a  counter-clockwise  direction,  and  will 
also  progress  in  a  train  as  suggested  by  the  large  arrow  Ay 
rolling  on  the  inner  surface  of  the  fixed  bearing,  represented 
by  the  portion  B.  From  this  it  will  be  seen  that  where  the 
balls  touch  5  and  B  the  friction  is  entirely  rolling  friction. 
But  if  we  consider  the  points  of  contact  between  the  balls 


I78 


THEORETICAL   MECHANICS. 


themselves  it  will  be  observed  that  at  all  these  points  there 
is  sliding  friction.  For  example,  in  Fig.  1 30,  which  represents 
on  a  larger  scale  the  adjacent  balls  b  and  bl  of  Fig.  129, 


Fig.  129. 

the  point  p  of  ball  b  is  shown  to  be  moving  upward,  and 
at  the  same  instant  the  point  pl  of  ball  bl  is  necessarily 
moving  downward,  if  the  two  balls  are  revolving  in  the 
same  direction. 


Fig.  130. 

The  occurrence  of  this  sliding  at  each  point  of  contact 
between  the  balls  is  sometimes  urged  as  an  argument  against 
the  use  of  ball  bearings,  notwithstanding  that  their  great 


FRICTION.  179 

efficiency  has  been  so  widely  demonstrated  in  practice. 
While  it  is  true  that  the  balls  must  slide  past  each  other,  it 
is  also  true  that  there  is  no  great  pressure  between  them  for 
the  reason  that  the  weight  on  the  axle  S  tends  to  .spread  the 
balls  outward  in  the  direction  of  the  radial  lines  shown  in 
Fig.  131,  thus  tending  to  separate  them  from  each  other  in 
the  manner  illustrated  by  the  small  dotted  circles  in  the 


Fig.  131. 


figure.  Hence,  in  the  absence  of  a  very  appreciable  pressure 
at  the  points  of  contact,  the  friction  between  the  balls  as 
they  slide  past  each  other  cannot  be  great. 

This  sliding  friction,  however,  such  as  it  is,  exists  at 
every  point  of  contact,  so  that  the  greater  the  number  of 
balls  used  the  greater  the  total  friction.  Hence  the  advantage 
of  using  a  small  number  of  large  balls,  rather  than  a  larger 
number  of  smaller  ones.  The  larger  balls  also  have  the 
advantage  that  their  tendency  to  be  pressed  away  from 
each  other  is  greater  because  of  the  greater  angle  ft, 
indicated  in  Fig.  131. 


l8o  THEORETICAL   MECHANICS. 

Lubricated  Surfaces  Depart  Widely  from  the  Ordinary 
Laws  of  Sliding  Friction,  When  two  surfaces  as  in  shaft 
bearings  are  separated  by  a  lubricant,  such  as  an  oil,  a 
grease,  or  plumbago,  the  simple  laws  of  sliding  friction 
are  greatly  modified,  especially  as  regards  kinetic  friction, 
or  friction  of  motion  The  friction  is  no  longer  between  the 
given  solid  surfaces  entirely,  or  under  some  circumstances 
at  all.  Much  depends  (i)  upon  the  lubricant  itself, — the 
kind  and  quantity,  and  the  manner  in  which  it  is  applied  ; 
and  (2)  upon  the  relative  velocity  of  the  two  surfaces,  and 
the  pressure  between  them.  (3)  A  temperature  change  at 
the  two  surfaces  is  also  sufficient  .to  change  the  entire 
relations  otherwise  established. 

Notice  that  these  are  not  simple  changes  in  the  values  of 
constant  coefficients,  but  are  radical  changes  of  fundamental 
laws,  giving  rise  to  variable  coefficients  for  the  same 
surfaces.  At  first  thought  one  would  say  that  if  the 
lubricant  adheres  to  each  of  the  two  rubbing  surfaces,  the 
motion  must  be  a  sliding  of  one  part  of  the  lubricant  on 
another  part,  and  hence  it  would  be  sufficient  to  substitute 
for  the  coefficient  of  friction  between  the  given  surfaces  the 
coefficient  of  oil  on  oil,  or  plumbago  on  plumbago,  etc. 
This  idea,  however,  is  not  upheld  by  careful  observations 
that  have  been  made. 

Scientific  investigation  has  not  furnished  altogether 
conclusive  results  as  to  the  laws  that  apply  to  the  friction  of 
lubricated  surfaces.  It  is  certain,  however,  that  the  friction 
under  these  circumstances  is  not  a  constant  ratio  of  the 
pressure  between  the  surfaces,  as  in  the  absence  of  lubrica- 
tion. This  ratio  (the  coefficient  of  friction)  for  lubricated 
surfaces  decreases  as  the  pressure  increases  and  continues  to 
do  so  up  to  a  certain  limit,  beyond  which  it  becomes  greater 
again  up  to  the  time  "cutting"  or  abrasion  begins. 


FRICTION.  iSl 

For  unlubricated  surfaces  the  difference  between  static 
and  kinetic  friction  (friction  to  start  the  motion  as  contrasted 
with  friction  during  the  motion)  is  never  very  great.  Between 
lubricated  surfaces  the  friction  at  starting  is  much  greater  in 
proportion, — until  the  motion  of  the  surfaces  has  carried  the 
lubricant  to  the  bearings,  from  which  it  has  been 
squeezed  out  during  rest.  This  is  especially  the  case  in  the 
bearings  of  heavy  machinery,  or  where  the  lubricant  is 
unduly  limpid. 

The  controlling  conditions — kind  of  lubricant,  its 
quantity  and  manner  of  application  ;  temperature  ;  velocity; 
and  pressure  in  bearings — are  inter-related  in  such  a 
complex  manner  that  it  is  practically  impossible  to  express 
these  relations  in  any  but  a  very  general  way  and  with  more 
than  approximate  correctness. 

In  general,  the  lubricant  should  adhere  to  the  surfaces 
sufficiently  to  be  constantly  dragged  in  to  the  rubbing  aiea 
as  fast  as  needed.  It  can  be  forced  in  by  external  pressure, 
under  some  circumstances. 

If  the  pressure  between  the  bearings  is  very  great  the 
the  lubricant  is  forced  out,  especially  if  it  is  too  thin,  and 
in  such  cases  it  is  better  to  use  grease  or  plumbago,  or 
"heavy"  oil.  In  small  bearings,  under  light  pressure,  a 
thick,  viscous  oil  would  be  needlessly  cohesive. 

Likewise,  at  varying  velocities  the  amount  of  lubricant 
drawn  into  the  rubbing  area  by  the  motion  of  the  bearings, 
and  the  general  effect,  is  very  variable.  At  moderate  speeds 
the  coefficient  of  friction  decreases  as  the  velocity  increases, 
but  for  very  high  speeds  the  opposite  is  true.  And 
furthermore,  the  speed  at  which  this  change  from  a 
decreasing  to  an  increasing  coefficient  takes  place  is 
different  for  high  and  low  pressures,  and  for  high  and  low 
temperatures  and  for  different  lubricants. 


1 82  THEORETICAL   MECHANICS. 

Conditions  that  are  satisfactory  at  one  temperature 
might  be  actually  reversed  at  even  a  slightly  different 
temperature.  The  properties  of  oils  change  greatly  under 
the  influence  of  heat,  and  no  two  oils  change  in  the  same 
manner  or  degree.  In  general,  the  lighter  oils  used  for 
high  speeds  and  low  pressures  give  better  service  at 
comparatively  high  temperatures,  while  the  more  viscous 
oils  and  greases  used  for  high  pressures  lose  their  desired 
viscosity  when  heated.  If  the  temperature  becomes  very 
high  the  organic  (animal  and  vegetable)  oils  may  be 
decomposed,  liberating  injurious  acid  compounds  that 
attack  the  material  of  the  bearings.  In  steam  chests  and 
cylinders  only  or  mainly  mineral  oils,  which  do  not  suffer 
decomposition  under  heat,  should  be  used. 

On  the  whole  the  friction  of  lubricated  surfaces  presents 
many  questions  for  scientific  investigation,  and  it  is  not  safe 
to  draw  very  broad  conclusions  from  the  superficial  and 
limited  observations  that  are  likely  to  be  met  with  in  this 
connection  in  ordinary  practice. 

• 

Friction  of  Ropes,  Belts  and  Cables.  When  a  flexible 
belt  or  cord  is  wrapped  upon  a  cylindrical  surface,  any 
tension  exerted  on  the  belt  or  cord  produces  a  normal 
pressure  against  the  cylindrical  surface  at  every  point  in  the 
arc  of  contact.  This  obviously  will  cause  friction,  and 
while  this  friction  will  be  in  accordance  with  the  general 
laws  already  deduced,  it  will  also  involve  conditions  and 
circumstances  requiring  special  consideration. 

The  hitherto  simple  law  of  sliding  friction  (F  =  <f>  P), 
when  extended  to  cover  the  case  of  belts  and  ropes  on 
pulleys  and  sheaves,  becomes  so  complex  that  it  cannot  be 
expressed  without  the  use  of  exponential  or  logarithmic 
formulae. 


FRICTION.  183 

Suppose  that  the  circle  C  in  Fig.  132,  represents  a 
cylinder  screwed  firmly  to  a  horizontal  board  so  that  it 
cannot  rotate*  A  cord  is  placed  around  the  cylinder  in  the 
manner  shown,  affording  a  contact  for  a  half  circumference, 
or  1 80°.  One  end  leads  to  a  spring  balance  fastened  to  the 
board.  Now  if  a  tension  T  be  cautiously  exerted  on  the 
free  end  of  the  cord  (avoiding  sudden  jerks)  it  will  be  found 
by  reading  the  spring  balance  that  the  tension  /  in  the  other 


Fig.  132. 

part  of  the  rope  is  much  less  than  T.  If  the  tension  T  be 
increased,  /  will  also  be  found  to  be  greater  in  the  same 
proportion.  That  is,  if  the  two  ends  of  the  cord  are  kept 
parallel  (so  as  not  to  change  the  length  of  the  arc  of  contact), 


the  ratio  —=  will 


remain   constant ;    t   will   always   be   a 

definite  portion  of  the  applied  tension  T.  Small  t  is  less 
than  T  because  of  the  friction  of  the  rope  or  belt  on  the 
cylinder,  and  this  friction  is  obviously  equal  to  the  difference 


*The  transmission  of  power  by  means  of  belts  and  ropes  presupposes  that  the 
pulley  moves  with  the  belt— a  sort  of  rolling  contact.  Some  fundamental 
conceptions,  however,  can  be  grasped  more  readily  by  first  regarding  the  pulley  as 
immovable.  In  fact  the  maximum  power  transmissible  is  determined  by  the 
conditions  at  the  time  of  slipping.  If  the  pulley  moves  with  the  belt  the  coefficient 
is  one  of  static  friction  ;  if  the  belt  is  intended  to  slip  on  the  pulley,  as  in  the 
friction  brake,  the  coefficient  is  for  kinetic  friction.  The  relations  above  deduced 
are  the  same  in  both  cases,  the  only  difference  being  in  the  magnitude  of  the 
constant  coefficients  for  static  and  kinetic  conditions. 


184  THEORETICAL   MECHANICS. 

T  —  /.  For  example,  if  T  —  5  pounds,  and  /  ==  i  pound, 
the  friction  of  the  cord  on  the  cylinder  is  F  =  4  pounds. 
If  we  increase  Tto  15  pounds,  then  t  will  be  3  pounds  and 
F  ~  T  —  t  =  12  pounds.  Expressed  in  different  form,  if 


Now  it  is  this  fact—  that  the  friction  is  itself  a 
function  of  the  force  that  overcomes  it—  which  gives  rise  to 
the  complex  mathematical  relations  that  have  to  be  dealt 
with  in  studying  the  friction  of  belts  and  ropes.  In  the 
simple  phenomenon  of  sliding  friction,  the  force  which 
overcomes  the  friction  between  the  weight  W  (Fig.  133)  and 


W 


Flg-  J33- 

the  surface  upon  which  it  rests  has  no  effect  to  change  the 
magnitude  of  this  friction  one  way  or  the  other.  In  the 
case  of  the  friction  of  the  rope  on  the  cylinder,  on  the 
contrary,  the  very  tension  that  is  intended  to  make  the  cord 
move  is  what  causes  the  friction  on  the  cylinder,  and  the 
greater  the  tension  the  greater  the  friction.  Hence,  in 
practice,  the  more  power  to  be  transmitted  the  tighter  the 
belt,  and  the  wider  and  thicker  it  must  be,  to  stand  the 
necessary  tension. 

The  Friction  of  a  Belt  Depends  upon  the  Are  of  Contact. 

The  friction  of  a  belt  on  a  pulley  is  not  only  a  function  of 
the  applied  tension,  but  depends  also  upon  the  magnitude  of 
the  arc  of  contact.  In  Fig.  132  the  rope  or  belt  was  in 
contact  with  one-half  the  circumference  of  the  pulley.  If 
now  the  end  to  which  the  tension  is  applied  has  a  direction 


FRICTION. 


185 


such  that  the  rope  is  in  contact  with  an  arc  ft,  less  than  the 
semi-circumference,  then  the  ratio  -=  is  no  longer  —  ,  asin 

-/  0 

the  last  paragraph,  nor  is  the  friction  F  -  -  —  T.     The  effect 

of  reducing  the  arc  of  contact  will  be  to  diminish  the 
friction,  leaving  a  greater  proportion  of  the  applied  tension, 
7",  to  be  propagated  along  the  cord  or  belt  to  the  spring 
balance,  to  cause  tension  t. 


Fig-  134. 

Hence,  the  friction  depends  upon  the  extent  to  which 
the  belt  envelops  the  pullej^.  This  is  not,  as  it  might 
appear,  in  violation  of  the  law  asserting  that  the  sliding 
friction  between  two  objects  is  independent  of  the  area  of 
contact.  The  friction  between  two  surfaces  is  independent 
of  the  area  of  contact,  provided  that  in  changing  the  area 
we  do  not  at  the  same  time  change  the  pressure  between  the 
surfaces.  For  instance,  if  a  brick  is  turned  from  a  flat  side 
to  an  edge,  the  area  of  the  surface  of  contact  is  changed, 
but  the  total  normal  pressure  is  still  the  weight  of  the  brick; 
the  area  is  less,  but  the  pressure  per  unit  area  is  greater. 
But  if  several  bricks  are  fastened  together  in  a  "train"  as  in 
Fig.  135,  the  pressure  per  unit  area  is  unchanged,  and 
hence  the  increased  area  of  contact  is  accompanied  by  an 


1 86  THEORETICAL   MECHANICS. 

increase  of  total  pressure,  thereby  multiplying  the  total 
friction  in  the  same  ratio.  The  friction  of  a  belt  on  a  pulley 
is  analogous  to  the  train  of  bricks.  Every  degree  added  to 
the  arc  of  contact  adds  to  the  total  pressure  of  the  belt 
against  the  pulley  and  thus  increases  the  friction. 


w 

— 

w 

w 

^ 

•^ 

Fig.  135- 

Friction  of  a  Flexible  Cord  Completely  Enveloping  a 
Cylindrical  Surface.  The  laws  for  the  friction  of  belts  and 
ropes  apply  not  only  to  ordinary  transmission  of  power  from 
pulley  to  pulley,  but  also  to  cases  where  a  rope  is  coiled 
several  times  around  a  cylinder,  as  the  rope  on  the  drum  of 
a  hoisting  engine  or  a  ship's  hawser  around  a  post.  If  there 
is  but  one  turn  of  the  rope  around  the  cylinder,  the  arc  of 
contact  is  360°,  for  two  turns  720°,  etc.  The  tension  that 
must  be  exerted  at  the  free  end  to  prevent  slipping  is,  as 
before,  the  difference  between  the  applied  tension  T  and  the 
friction.  With  this  same  applied  tension,  if  we  take  a 
second  turn  of  the  rope  around  the  cylinder,  the  value  t 
left  after  one  turn  may  be  regarded  as  the  "applied  tension" 
for  the  next  turn,  so  that  if  /.,  is  the  tension  that  remains  in 
-the  free  end  after  two  turns,  then 


If  there  were  three  turns  of  the  rope  the  tension  in  the  free 
end,  ^3,  would  be  found  from  the  relation, 


t2     ~   t         T'         T      \  T 

For    example,    fasten    a    cylinder    of    3-inch   or   4-inch 
diameter  to  a  board,   and  screw  this  board  to  the  wall.     A 


FRICTION. 


I87 


stout    cord  is   substituted  for  a  rope.     The   upper   end  of 
the  cord  is  hooked  onto  a  sensitive  spring  balance,  and  the 


The   ratio  -= 


applied    tension    is   produced   by  weights. 

for  one  turn  is  nearly  constant  for  all 
tensions,  but  not  absolutely  so,  because 
on  account  of  the  twisting  of  the  strands 
the  flexibility  of  the  rope  is  different  for 
different  values  of  T. 

roughly,  —  =  -1 

second  turn  around  the  post  this  value, 

T 

t  or  — - ,  must  be  taken  as  the  applied 

tension  for   the  second  turn,  and  the 
tension  on  the  balance    will     now   be 
i 


Suppose  that, 
Now  if  we  take  a 


(approximately)   -  —  /or—      71 
6  36 

three  turns  the  balance  will  read 

T  T  T 

7. 


For 


In  general  terms,  if  /n  is  the  tension  left  after  n  turns  of  the 
rope,  then  <- 

' 


where  the  exponent  n  is  the  number  of  turns,   and  K  the 
constant  depending  on  the  nature  of  the  surfaces. 

This  is  sufficient  to  show  in  a  general  way  that  the 
various  factors  to  be  considered  in  the  friction  of  belts  and 
ropes  are  related  to  each  other  in  such  a  wray  as  to  give  rise 
to  mathematical  expressions  of  an  exponential  character. 
In  the  next  paragraph  these  relations  are  expressed  by 
means  of  logarithms  in  a  usable  form  for  the  ordinary 
practice  of  belting. 


1 88 


THEORETICAL   MECHANICS. 


Mathematical  Expression  of  the  Friction  of  a  Belt  or 
Rope  in  Terms  of  the  Tensions  and  the  Are  of  Contact. 
Having  shown  in  a  general  way  that  the  friction  of  a  belt  or 

rope  on  its  pulley  depends  upon 

(1)  the  tension  of  the  belt,   and 

(2)  the  magnitude  of  the  arc  of 
contact,  it  may  be  well  to  express 
these    relations   in   exact   math- 
ematical form.     We  can  only  out- 
line the  method  of  deducing  these 
expressions.* 

Suppose  that  ^C(Fig.  137) 
represents  a  belt  bent  at  an  angle 
ft  over  the  edge  of  a  block  of 
iron,  the  angle  (3  being  deter- 
mined by  its  supplementary  angle 
BCA  in  the  block.  The  applied 
tension  T  is  sufficient  to  over- 


Fig.  137- 


come  the  friction  necessary  to  slide  the  belt  over  the  angle 
at  C  and  to  leave  a  tension  /  in  the  other  end  of  the  rope, 
or  T  -  F  -\-  t,  where  F  represents  the  friction  at  C.  The 
friction  at  C  is  caused {  of  course,  by  the  normal  pressure  of 
the  belt  against  the  block  at  that  point,  and  this  normal 
pressure  is  merely  the  resultant,  R,  of  T  and  t.  There  is  no 
friction  on  the  faces  of  the  block  in  contact  with  the  belt, 
because  there  is  no  pressure  of  the  belt  against  those  faces. 
Hence  we  have  F  -  $  R.  But 


whence 


R  =  V  T2  +  t2    -  2  Tt  cos  /3] 
F=$  \f  T2  +  t2  —  2  TV  cos  0. 


*For  full  demonstration  see  Cromwell's  "Belts  and  Pulleys",  Section  X, 
—Wiley  &  Sons,  New  York,  Publishers. 

**See  Kinematics,  p.  14.  Notice  the  negative  sign  of  the  third  term  under 
the  radical  sign,— due  to  the  fact  that  BAC\s  180°  —  /3. 


FRICTION. 


189 


Using  this  equation  and  the  relation  F  -  T  —  t,  by  proper 
algebraic  and  trigonometric  transformations,  we  may  deduce 
the  expression 

T 


sn 


ft 


Fig.  138. 

Now  what  is  true  of  this  ft  will  be  true  in  a  general  way 
for  each  of  the  angles,  ft,  y,  8,  etc.  in  Fig.  138. 

From  a  large  number  of  finite 
angles  we  can  reason  to  an 
infinite  number  of  infinitesmal 
angles,  such  as  would  constitute 
the  total  angle  ft  in  Fig.  139, 
illustrating  a  belt  and  pulley. 
By  this  process  we  would 
arrive  at  the  following  general 
formula,  representing  the  rela- 
tions between  the  tensions  7~and 
/,  and  the  angle  ft,  and  expressed 
Fig.  139.  as  a  common  logarithm: 


IQO  THEORETICAL   MECHANICS. 

log  —  =  0.007578  0  p, 
in  which 

ft   is  expressed  in  degrees  ; 

<j>  is  the  coefficient  of  friction  ; 

T  is  the  tension  of  the  tight  side  ;  and 

t    is  the  tension  of  the  loose  side. 

From  this  formula  we  can  deduce  an  expression  showing 
directly  the  relation  between  the  friction,  F\  the  applied 
tension,  T;  the  arc  of  contact,  ft',  and  the  coefficient  of 
friction  Expressing  the  above  formula  as  an  anti-logarithm, 
we  get  the  expression 


or,  inverting, 


—  =   log -1 0.0075^8  0  p  ' 


T     '    Iog~1o.oo757«  0/3  ' 


Now  remembering  that  /  is  that  part  of  T  which  is  left  over 
after  overcoming  the  friction,  and  putting  the  expression 
F  =  T  —  /in  the  form 


and  substituting  for        the  value  deduced  above,  we  get 


To  use  this  formula  in  practice,  it  is  necessary  to  know  the 
coefficient  of  friction  for  the  various  kinds  of  ropes,  belts  and 
pulleys  in  common  use. 


FRICTION.  IQI 

For  leather  belts  on  iron  pulleys,  <£  =  0.40 

For  leather  belts  on  leather-covered  pulleys,  <f>  =  O-45- 

For   hemp  rope  on  iron  in  semi-circular  groove  (Fig. 

140),  </>  =  0.30.     (Used  for  small  powers  only) . 

For  hemp  rope  on  iron  in  V-groove,  with  straight  sides 

inclined  at  angle  of  45°  (Fig.  141),   $  =  0.70. 

For  steel  cable  in  groove  lined  with  leather  (Fig.  142), 

(f>  =  0.24. 


Fig.  140 


Fig.  141. 


Fig.  142. 


Examples : 

1.  A  leather  belt  runs  over  a  cast  iron  pulley  enveloping  an  arc 
of  140°.     What  is  the  friction   between   the  belt  and  the  pulley  if 
slipping  begins   when   the  tension  in   the  driving  side  of  the  belt  is 
go  pounds  ? 

2.  A  rope  belt  running  over  a  pulley  in  a  V r -groove  has  an  arc  of 
contact  of  180°.     A  force  of  200  pounds  applied  to  the  perimeter  of  the 
wheel  is  just  sufficient  to  prevent  it  from  rotating  so  that  the  rope  has 
to  slip  over  the  pulley.     What  is  the  value  of  T  ? 

3.  A    leather  belt  envelops  an  arc  of  ijo°  on  a  leather-covered 
pulley.     What  will  be  the  friction  when  T  —.  200  pounds  f 

4.  The  friction  of  a  leather  belt  enveloping  an  arc  of  190°  on  a 
leather-covered  pulley  is   //j  pounds.       What  is  the   tension  in  the 
driving  side  of  the  belt,  and  what  is  the  tension  in  the  slack  slide  f 


1 92  THEORETICAL   MECHANICS. 

Measurement  of  Power  Transmitted  by  Belts.  In  the 
ordinary  continuous  transmission  of  power  by  means  of  an 
endless  belt,  the  angle  of  contact  on  each  of  the  two  pulleys 
depends  upon  their  relative  diameters  and  their  distance 
apart.  The  maximum  power  that  can  be  transmitted  from 
one  pulley  to  the  other  through  the  belt  depends  upon  the 
friction  at  that  pulley  where  slipping  will  first  occur.  If 
both  pulleys  are  of  the  same  material  this  will  be  at  the 
pulley  having  the  smaller  arc  of  contact. 


In  Fig.  143,  illustrating  pulleys  of  2o-inch  and  lo-inch 
diameters  respectively,  if  pl  is  the  driving  pulley,  it  will  not 
be  possible  for  the  belt  to  transmit  to  the  second  pulley,  p^ 
all  the  power  that  the  driving  pulley  would  be  capable  of 
giving  to  the  belt.  The  friction  of  the  belt  on  pt>,  in 
causing  the  motion  of  the  pulley,  is  equivalent  to  a  force  of 
the  same  magnitude  acting  at  any  single  point  on  the 
perimeter  of  p 2  in  a  tangent  direction.  And  it  should  be 
remembered,  again,  that  this  maximum  force  is  not  T,  the 
tension  in  the  belt,  but  only  a  part  of  it, — that  is,  the 
friction  /  or 


\l     ~ 


log 


FRICTION.  193 

If  the  diameter  of  p.£  is  d  inches,  then  for  each  revolution 
of  pi  the  maximum  work  that  can  be  done,  or  the  maximum 
energy  that  can  be  taken  from  the  belt  by  this  pulley,  is 

trd 

F  X  —foot-pounds  (provided  F  is  in  pounds). 

If  this  pulley  has  n  revolutions  per  minute,  the 
maximum  H.  P.  that  can  be  transmitted  to  it  from  the 
belt  is 


12        33  >000 
Examples : 

\.  In  the  case  of  the  pulleys  and  endless  belt  just  referred  to, 
if  pi  and  p2  have  diameters  of  20  inches  and  10  inches  respectively, 
and  are  12  feet  apart  between  centers,  what  is  the  magnitude  of  the 
arc  of  contact  of  the  belt  on  each  of  the  pulleys  ? 

2.  If  the  shaft  of  ^^  revolves  165  times  per  minute,  what  is  the 
maximum  power  that  can  be  conveyed  to  the  shaft  of  p.,,  if  the  tension 
of  the  driving  side  of  the  belt  is  211  pounds  when  slipping  commences? 


Methods  of  Increasing  the  Efficiency  of  Belts.  Anything 
that  will  increase  the  tension  of  the  belt,  or  the  arc  of 
contact,  or  the  coefficient  of  friction,  will  increase  the 
maximum  power  transmissible  by  the  belt. 

We  have  already  seen  that  the  coefficient  is  increased 
from  0.4  to  0.45  by  covering  the  pulley  with  leather.  The 
two  pulleys  should  not  be  near  enough  together  to  interfere 
with  the  flexible  working  of  the  belt.  If  the  pulleys  are  of 
unequal  size  the  arc  of  contact  on  the  smaller  will  be 
greater,  the  farther  apart  the  pulleys.  If  the  slack  side  of 
the  belt  is  uppermost,  the  sag  of  the  belt  will  increase  the 
arcs  of  contact. 


THEORETICAL    MECHANICS. 


By  the  use  of  a  so-called  "idle"  or  tightening  pulley,  as 
illustrated  in  Figs.'  144  and   145,  it  is  possible  to  increase 


Fig-  US- 


both  the  tension  and  the  arcs  of  contact  at  the  same  time. 
Fig.  144  illustrates  the  method  of  using  a  tightening  pulley 
on  horizontal  belts  ;  Fig.  145  is  for  vertical  belts. 

Favorable  results  have  been  reported  of  a  simple  and 
apparently  practical  method  of  increasing  the  normal 
pressure  between  the  belt  and  the  pulley  without  subjecting 
the  belt  to  excessive  tension.  A  strip  of  canvas  (or  other 
suitable  material)  narrower  than  the  belt,  together  with  a 
short  length  of  stout  sheet  rubber  to  be  inserted  between 
the  ends  of  the  canvas  strip,  is  made  into  a  band.  This  band 
is  stretched  garter-like  over  the  belt. 


FRICTION.  195 

With  a  little  thought  it  will  be  seen  that  the  action  of 
any  belt  necessitates  some  slipping.  One  side  of  the  belt 
being  tighter  than  the  other,  as  each  minute  portion  of  the 
belt  progresses  along  the  perimeter  of  the  driven  pulley  to  a 
position  of  greater  tension,  it  is  gradually  stretched  out,  and 
hence  must  slip  a  little.  As  it  again  approaches  the  slack 
side  of  the  other  pulley — the  driving  pulley — it  contracts 
and  slips  in  the  opposite  direction.  The  elasticity  of  the 
material,  therefore,  is  a  factor  to  be  considered  in  the  easy 
working  of  the  belt.  This  constant  slipping  generates  heat, 
causing  a  gradual  burning  of  the  belt. 

It  will  also  be  observed  from  formula  n,  p.  190,  that 
the  friction  depends  on  T,  <£  and  /3,  and  not  upon  the  width 
of  the  belt  or  diameter  of  the  pulley.  This  applies  down  to 
the  limit  at  which  the  lack  of  perfect  flexibility  in  the  belt 
would  prevent  it  from  conforming  properly  to  the  surface  of 
too  small  a  pulley. 


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